linear_algebra.free_module.pid ⟷ Mathlib.LinearAlgebra.FreeModule.PID

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -3,7 +3,7 @@ Copyright (c) 2020 Anne Baanen. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 -/
-import LinearAlgebra.Dimension
+import LinearAlgebra.Dimension.Basic
 import LinearAlgebra.FreeModule.Basic
 import RingTheory.PrincipalIdealDomain
 import RingTheory.Finiteness
@@ -376,7 +376,7 @@ noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι
 
 variable {M}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i «expr ∉ » I) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:642:2: warning: expanding binder collection (i «expr ∉ » I) -/
 #print Module.basisOfFiniteTypeTorsionFree /-
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -72,7 +72,7 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
   rw [Submodule.eq_bot_iff]
   intro x hx
   refine' b.ext_elem fun i => _
-  rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ 
+  rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
   rw [LinearEquiv.map_zero, Finsupp.zero_apply]
   exact
     (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _
@@ -88,7 +88,7 @@ theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M}
   rw [Submodule.eq_bot_iff]
   intro x hx
   refine' congr_arg coe (show (⟨x, hNO hx⟩ : O) = 0 from b.ext_elem fun i => _)
-  rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ 
+  rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
   rw [LinearEquiv.map_zero, Finsupp.zero_apply]
   refine' (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _ _
   exact (LinearMap.mem_submoduleImage_of_le hNO).mpr ⟨x, hx, rfl⟩
@@ -259,7 +259,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
     by
     intro c x xM' hc
     obtain ⟨⟨x, xM⟩, hx', rfl⟩ := submodule.mem_map.mp xM'
-    rw [LinearMap.mem_ker] at hx' 
+    rw [LinearMap.mem_ker] at hx'
     have hc' : (c • ⟨y', y'M⟩ + ⟨x, xM⟩ : M) = 0 := Subtype.coe_injective hc
     simpa only [LinearMap.map_add, LinearMap.map_zero, LinearMap.map_smul, smul_eq_mul, add_zero,
       mul_eq_zero, ϕy'_ne_zero, hx', or_false_iff] using congr_arg ϕ hc'
@@ -274,7 +274,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   · refine' Basis.mkFinConsOfLE y yN bN' N'_le_N _ _
     · intro c z zN' hc
       refine' ay'_ortho_N' c z zN' _
-      rwa [← a_smul_y'] at hc 
+      rwa [← a_smul_y'] at hc
     · intro z zN
       obtain ⟨b, hb⟩ : _ ∣ ϕ ⟨z, N_le_M zN⟩ := generator_submodule_image_dvd_of_mem N_le_M ϕ zN
       refine' ⟨-b, submodule.mem_map.mpr ⟨⟨_, N.sub_mem zN (N.smul_mem b yN)⟩, _, _⟩⟩
@@ -692,7 +692,7 @@ theorem LinearIndependent.restrict_scalars_algebras {R S M ι : Type _} [CommSem
     [AddCommMonoid M] [Algebra R S] [Module R M] [Module S M] [IsScalarTower R S M]
     (hinj : Function.Injective (algebraMap R S)) {v : ι → M} (li : LinearIndependent S v) :
     LinearIndependent R v :=
-  LinearIndependent.restrict_scalars (by rwa [Algebra.algebraMap_eq_smul_one'] at hinj ) li
+  LinearIndependent.restrict_scalars (by rwa [Algebra.algebraMap_eq_smul_one'] at hinj) li
 #align linear_independent.restrict_scalars_algebras LinearIndependent.restrict_scalars_algebras
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -382,19 +382,58 @@ variable {M}
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
     (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σ n : ℕ, Basis (Fin n) R M := by
   classical
+  -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
+  have := exists_maximal_independent R s
+  let I : Set ι := this.some
+  obtain
+    ⟨indepI : LinearIndependent R (s ∘ coe : I → M), hI :
+      ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I)⟩ :=
+    this.some_spec
+  let N := span R (range <| (s ∘ coe : I → M))
+  -- same as `span R (s '' I)` but more convenient
+  let sI : I → N := fun i => ⟨s i.1, subset_span (mem_range_self i)⟩
+  -- `s` restricted to `I`
+  let sI_basis : Basis I R N
+  -- `s` restricted to `I` is a basis of `N`
+  exact Basis.span indepI
+  -- Our first goal is to build `A ≠ 0` such that `A • M ⊆ N`
+  have exists_a : ∀ i : ι, ∃ a : R, a ≠ 0 ∧ a • s i ∈ N :=
+    by
+    intro i
+    by_cases hi : i ∈ I
+    · use 1, zero_ne_one.symm
+      rw [one_smul]
+      exact subset_span (mem_range_self (⟨i, hi⟩ : I))
+    · simpa [image_eq_range s I] using hI i hi
+  choose a ha ha' using exists_a
+  let A := ∏ i, a i
+  have hA : A ≠ 0 := by
+    rw [Finset.prod_ne_zero_iff]
+    simpa using ha
+  -- `M ≃ A • M` because `M` is torsion free and `A ≠ 0`
+  let φ : M →ₗ[R] M := LinearMap.lsmul R M A
+  have : φ.ker = ⊥ := LinearMap.ker_lsmul hA
+  let ψ : M ≃ₗ[R] φ.range := LinearEquiv.ofInjective φ (linear_map.ker_eq_bot.mp this)
+  have : φ.range ≤ N :=
+    by
+    -- as announced, `A • M ⊆ N`
+    suffices ∀ i, φ (s i) ∈ N
+      by
+      rw [LinearMap.range_eq_map, ← hs, φ.map_span_le]
+      rintro _ ⟨i, rfl⟩; apply this
+    intro i
+    calc
+      (∏ j, a j) • s i = (∏ j in {i}ᶜ, a j) • a i • s i := by
+        rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]
+      _ ∈ N := N.smul_mem _ (ha' i)
+  -- Since a submodule of a free `R`-module is free, we get that `A • M` is free
+  obtain ⟨n, b : Basis (Fin n) R φ.range⟩ := Submodule.basisOfPidOfLE this sI_basis
+  -- hence `M` is free.
+  exact ⟨n, b.map ψ.symm⟩
 #align module.basis_of_finite_type_torsion_free Module.basisOfFiniteTypeTorsionFree
 -/
 
 #print Module.free_of_finite_type_torsion_free /-
--- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
--- same as `span R (s '' I)` but more convenient
--- `s` restricted to `I`
--- `s` restricted to `I` is a basis of `N`
--- Our first goal is to build `A ≠ 0` such that `A • M ⊆ N`
--- `M ≃ A • M` because `M` is torsion free and `A ≠ 0`
--- as announced, `A • M ⊆ N`
--- Since a submodule of a free `R`-module is free, we get that `A • M` is free
--- hence `M` is free.
 theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs : span R (range s) = ⊤)
     [NoZeroSMulDivisors R M] : Module.Free R M :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -382,58 +382,19 @@ variable {M}
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
     (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σ n : ℕ, Basis (Fin n) R M := by
   classical
-  -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
-  have := exists_maximal_independent R s
-  let I : Set ι := this.some
-  obtain
-    ⟨indepI : LinearIndependent R (s ∘ coe : I → M), hI :
-      ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I)⟩ :=
-    this.some_spec
-  let N := span R (range <| (s ∘ coe : I → M))
-  -- same as `span R (s '' I)` but more convenient
-  let sI : I → N := fun i => ⟨s i.1, subset_span (mem_range_self i)⟩
-  -- `s` restricted to `I`
-  let sI_basis : Basis I R N
-  -- `s` restricted to `I` is a basis of `N`
-  exact Basis.span indepI
-  -- Our first goal is to build `A ≠ 0` such that `A • M ⊆ N`
-  have exists_a : ∀ i : ι, ∃ a : R, a ≠ 0 ∧ a • s i ∈ N :=
-    by
-    intro i
-    by_cases hi : i ∈ I
-    · use 1, zero_ne_one.symm
-      rw [one_smul]
-      exact subset_span (mem_range_self (⟨i, hi⟩ : I))
-    · simpa [image_eq_range s I] using hI i hi
-  choose a ha ha' using exists_a
-  let A := ∏ i, a i
-  have hA : A ≠ 0 := by
-    rw [Finset.prod_ne_zero_iff]
-    simpa using ha
-  -- `M ≃ A • M` because `M` is torsion free and `A ≠ 0`
-  let φ : M →ₗ[R] M := LinearMap.lsmul R M A
-  have : φ.ker = ⊥ := LinearMap.ker_lsmul hA
-  let ψ : M ≃ₗ[R] φ.range := LinearEquiv.ofInjective φ (linear_map.ker_eq_bot.mp this)
-  have : φ.range ≤ N :=
-    by
-    -- as announced, `A • M ⊆ N`
-    suffices ∀ i, φ (s i) ∈ N
-      by
-      rw [LinearMap.range_eq_map, ← hs, φ.map_span_le]
-      rintro _ ⟨i, rfl⟩; apply this
-    intro i
-    calc
-      (∏ j, a j) • s i = (∏ j in {i}ᶜ, a j) • a i • s i := by
-        rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]
-      _ ∈ N := N.smul_mem _ (ha' i)
-  -- Since a submodule of a free `R`-module is free, we get that `A • M` is free
-  obtain ⟨n, b : Basis (Fin n) R φ.range⟩ := Submodule.basisOfPidOfLE this sI_basis
-  -- hence `M` is free.
-  exact ⟨n, b.map ψ.symm⟩
 #align module.basis_of_finite_type_torsion_free Module.basisOfFiniteTypeTorsionFree
 -/
 
 #print Module.free_of_finite_type_torsion_free /-
+-- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
+-- same as `span R (s '' I)` but more convenient
+-- `s` restricted to `I`
+-- `s` restricted to `I` is a basis of `N`
+-- Our first goal is to build `A ≠ 0` such that `A • M ⊆ N`
+-- `M ≃ A • M` because `M` is torsion free and `A ≠ 0`
+-- as announced, `A • M ⊆ N`
+-- Since a submodule of a free `R`-module is free, we get that `A • M` is free
+-- hence `M` is free.
 theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs : span R (range s) = ⊤)
     [NoZeroSMulDivisors R M] : Module.Free R M :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -225,7 +225,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   let y' : O := ∑ i, c i • b'M i
   have y'M : y' ∈ M := M.sum_mem fun i _ => M.smul_mem (c i) (b'M i).2
   have mk_y' : (⟨y', y'M⟩ : M) = ∑ i, c i • b'M i :=
-    Subtype.ext (show y' = M.subtype _ by simp only [LinearMap.map_sum, LinearMap.map_smul]; rfl)
+    Subtype.ext (show y' = M.subtype _ by simp only [map_sum, LinearMap.map_smul]; rfl)
   have a_smul_y' : a • y' = y :=
     by
     refine' congr_arg coe (show (a • ⟨y', y'M⟩ : M) = ⟨y, N_le_M yN⟩ from _)

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,10 +3,10 @@ Copyright (c) 2020 Anne Baanen. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 -/
-import Mathbin.LinearAlgebra.Dimension
-import Mathbin.LinearAlgebra.FreeModule.Basic
-import Mathbin.RingTheory.PrincipalIdealDomain
-import Mathbin.RingTheory.Finiteness
+import LinearAlgebra.Dimension
+import LinearAlgebra.FreeModule.Basic
+import RingTheory.PrincipalIdealDomain
+import RingTheory.Finiteness
 
 #align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
 
@@ -376,7 +376,7 @@ noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι
 
 variable {M}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:641:2: warning: expanding binder collection (i «expr ∉ » I) -/
 #print Module.basisOfFiniteTypeTorsionFree /-
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}

mathlib commit https://github.com/leanprover-community/mathlib/commit/32a7e535287f9c73f2e4d2aef306a39190f0b504

@@ -271,7 +271,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
     rw [mul_comm, mul_smul, hc]
   -- So we can extend a basis for `N'` with `y`
   refine' ⟨y'_ortho_M', ay'_ortho_N', fun n' bN' => ⟨_, _⟩⟩
-  · refine' Basis.mkFinConsOfLe y yN bN' N'_le_N _ _
+  · refine' Basis.mkFinConsOfLE y yN bN' N'_le_N _ _
     · intro c z zN' hc
       refine' ay'_ortho_N' c z zN' _
       rwa [← a_smul_y'] at hc 
@@ -284,7 +284,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   -- And extend a basis for `M'` with `y'`
   intro m' hn'm' bM'
   refine' ⟨Nat.succ_le_succ hn'm', _, _⟩
-  · refine' Basis.mkFinConsOfLe y' y'M bM' M'_le_M y'_ortho_M' _
+  · refine' Basis.mkFinConsOfLE y' y'M bM' M'_le_M y'_ortho_M' _
     intro z zM
     refine' ⟨-ϕ ⟨z, zM⟩, ⟨⟨z, zM⟩ - ϕ ⟨z, zM⟩ • ⟨y', y'M⟩, linear_map.mem_ker.mpr _, _⟩⟩
     · rw [LinearMap.map_sub, LinearMap.map_smul, ϕy'_eq, smul_eq_mul, mul_one, sub_self]
@@ -293,7 +293,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   intro as h
   refine' ⟨Fin.cons a as, _⟩
   intro i
-  rw [Basis.coe_mkFinConsOfLe, Basis.coe_mkFinConsOfLe]
+  rw [Basis.coe_mkFinConsOfLE, Basis.coe_mkFinConsOfLE]
   refine' Fin.cases _ (fun i => _) i
   · simp only [Fin.cons_zero, Fin.castLE_zero]
     exact a_smul_y'.symm

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,17 +2,14 @@
 Copyright (c) 2020 Anne Baanen. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
-
-! This file was ported from Lean 3 source module linear_algebra.free_module.pid
-! leanprover-community/mathlib commit d87199d51218d36a0a42c66c82d147b5a7ff87b3
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.LinearAlgebra.Dimension
 import Mathbin.LinearAlgebra.FreeModule.Basic
 import Mathbin.RingTheory.PrincipalIdealDomain
 import Mathbin.RingTheory.Finiteness
 
+#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
+
 /-! # Free modules over PID
 
 > THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
@@ -379,7 +376,7 @@ noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι
 
 variable {M}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i «expr ∉ » I) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
 #print Module.basisOfFiniteTypeTorsionFree /-
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}

mathlib commit https://github.com/leanprover-community/mathlib/commit/2fe465deb81bcd7ccafa065bb686888a82f15372

@@ -193,9 +193,9 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
                   ∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
                     ∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
                       ∀ (as : Fin n' → R)
-                        (h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)),
+                        (h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLEEmb hn'm' i) : O)),
                         ∃ as' : Fin (n' + 1) → R,
-                          ∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O) :=
+                          ∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLEEmb hnm i) : O) :=
   by
   -- Let `ϕ` be a maximal projection of `M` onto `R`, in the sense that there is
   -- no `ψ` whose image of `N` is larger than `ϕ`'s image of `N`.
@@ -492,7 +492,7 @@ This is a strengthening of `submodule.basis_of_pid_of_le`.
 theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M) (N O : Submodule R M)
     (N_le_O : N ≤ O) :
     ∃ (n o : ℕ) (hno : n ≤ o) (bO : Basis (Fin o) R O) (bN : Basis (Fin n) R N) (a : Fin n → R),
-      ∀ i, (bN i : M) = a i • bO (Fin.castLE hno i) :=
+      ∀ i, (bN i : M) = a i • bO (Fin.castLEEmb hno i) :=
   by
   cases nonempty_fintype ι
   revert N
@@ -527,7 +527,7 @@ noncomputable def Submodule.smithNormalFormOfLE [Finite ι] (b : Basis ι R M) (
   by
   choose n o hno bO bN a snf using N.exists_smith_normal_form_of_le b O N_le_O
   refine'
-    ⟨o, n, bO, bN.map (comap_subtype_equiv_of_le N_le_O).symm, (Fin.castLE hno).toEmbedding, a,
+    ⟨o, n, bO, bN.map (comap_subtype_equiv_of_le N_le_O).symm, (Fin.castLEEmb hno).toEmbedding, a,
       fun i => _⟩
   ext
   simp only [snf, Basis.map_apply, Submodule.comapSubtypeEquivOfLe_symm_apply,

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -67,6 +67,7 @@ variable {ι : Type _} (b : Basis ι R M)
 
 open Submodule.IsPrincipal Submodule
 
+#print eq_bot_of_generator_maximal_map_eq_zero /-
 theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
     {ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
     (hgen : generator (N.map ϕ) = (0 : R)) : N = ⊥ :=
@@ -80,7 +81,9 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
     (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _
       ⟨x, hx, rfl⟩
 #align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zero
+-/
 
+#print eq_bot_of_generator_maximal_submoduleImage_eq_zero /-
 theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
     (hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (hgen : generator (ϕ.submoduleImage N) = 0) : N = ⊥ :=
@@ -93,6 +96,7 @@ theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M}
   refine' (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _ _
   exact (LinearMap.mem_submoduleImage_of_le hNO).mpr ⟨x, hx, rfl⟩
 #align eq_bot_of_generator_maximal_submodule_image_eq_zero eq_bot_of_generator_maximal_submoduleImage_eq_zero
+-/
 
 end Ring
 
@@ -126,6 +130,7 @@ variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal
 
+#print generator_maximal_submoduleImage_dvd /-
 theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
     (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (y : M) (yN : y ∈ N)
@@ -161,7 +166,9 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
   exact ideal.span_singleton_le_span_singleton.mpr d_dvd_left
   · exact subset_span (mem_insert _ _)
 #align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvd
+-/
 
+#print Submodule.basis_of_pid_aux /-
 /-- The induction hypothesis of `submodule.basis_of_pid` and `submodule.smith_normal_form`.
 
 Basically, it says: let `N ≤ M` be a pair of submodules, then we can find a pair of
@@ -295,7 +302,9 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
     exact a_smul_y'.symm
   · rw [Fin.castLE_succ]; simp only [Fin.cons_succ, coe_of_le, h i]
 #align submodule.basis_of_pid_aux Submodule.basis_of_pid_aux
+-/
 
+#print Submodule.nonempty_basis_of_pid /-
 /-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
 
@@ -320,6 +329,7 @@ theorem Submodule.nonempty_basis_of_pid {ι : Type _} [Finite ι] (b : Basis ι
   obtain ⟨bN, hbN⟩ := h' n' bN'
   exact ⟨n' + 1, ⟨bN⟩⟩
 #align submodule.nonempty_basis_of_pid Submodule.nonempty_basis_of_pid
+-/
 
 #print Submodule.basisOfPid /-
 /-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
@@ -333,6 +343,7 @@ noncomputable def Submodule.basisOfPid {ι : Type _} [Finite ι] (b : Basis ι R
 #align submodule.basis_of_pid Submodule.basisOfPid
 -/
 
+#print Submodule.basisOfPid_bot /-
 theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
     Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ :=
   by
@@ -341,7 +352,9 @@ theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
   obtain rfl : n = 0 := by simpa using fintype.card_eq.mpr ⟨e⟩
   exact Sigma.eq rfl (Basis.eq_of_apply_eq <| finZeroElim)
 #align submodule.basis_of_pid_bot Submodule.basisOfPid_bot
+-/
 
+#print Submodule.basisOfPidOfLE /-
 /-- A submodule inside a free `R`-submodule of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
 
@@ -352,7 +365,9 @@ noncomputable def Submodule.basisOfPidOfLE {ι : Type _} [Finite ι] {N O : Subm
   let ⟨n, bN'⟩ := Submodule.basisOfPid b (N.comap O.Subtype)
   ⟨n, bN'.map (Submodule.comapSubtypeEquivOfLe hNO)⟩
 #align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLE
+-/
 
+#print Submodule.basisOfPidOfLESpan /-
 /-- A submodule inside the span of a linear independent family is a free `R`-module of finite rank,
 if `R` is a principal ideal domain. -/
 noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι → M}
@@ -360,10 +375,12 @@ noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι
     Σ n : ℕ, Basis (Fin n) R N :=
   Submodule.basisOfPidOfLE le (Basis.span hb)
 #align submodule.basis_of_pid_of_le_span Submodule.basisOfPidOfLESpan
+-/
 
 variable {M}
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i «expr ∉ » I) -/
+#print Module.basisOfFiniteTypeTorsionFree /-
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
     (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σ n : ℕ, Basis (Fin n) R M := by
@@ -417,7 +434,9 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M
   -- hence `M` is free.
   exact ⟨n, b.map ψ.symm⟩
 #align module.basis_of_finite_type_torsion_free Module.basisOfFiniteTypeTorsionFree
+-/
 
+#print Module.free_of_finite_type_torsion_free /-
 theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs : span R (range s) = ⊤)
     [NoZeroSMulDivisors R M] : Module.Free R M :=
   by
@@ -425,22 +444,28 @@ theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs :
   obtain ⟨n, b⟩ : Σ n, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree hs
   exact Module.Free.of_basis b
 #align module.free_of_finite_type_torsion_free Module.free_of_finite_type_torsion_free
+-/
 
+#print Module.basisOfFiniteTypeTorsionFree' /-
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree' [Module.Finite R M]
     [NoZeroSMulDivisors R M] : Σ n : ℕ, Basis (Fin n) R M :=
   Module.basisOfFiniteTypeTorsionFree Module.Finite.exists_fin.choose_spec.choose_spec
 #align module.basis_of_finite_type_torsion_free' Module.basisOfFiniteTypeTorsionFree'
+-/
 
+#print Module.free_of_finite_type_torsion_free' /-
 theorem Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMulDivisors R M] :
     Module.Free R M :=
   by
   obtain ⟨n, b⟩ : Σ n, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree'
   exact Module.Free.of_basis b
 #align module.free_of_finite_type_torsion_free' Module.free_of_finite_type_torsion_free'
+-/
 
 section SmithNormal
 
+#print Basis.SmithNormalForm /-
 /-- A Smith normal form basis for a submodule `N` of a module `M` consists of
 bases for `M` and `N` such that the inclusion map `N → M` can be written as a
 (rectangular) matrix with `a` along the diagonal: in Smith normal form. -/
@@ -452,7 +477,9 @@ structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type _) (n : ℕ) wher
   a : Fin n → R
   snf : ∀ i, (bN i : M) = a i • bM (f i)
 #align basis.smith_normal_form Basis.SmithNormalForm
+-/
 
+#print Submodule.exists_smith_normal_form_of_le /-
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
@@ -483,7 +510,9 @@ theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M)
   obtain ⟨as, has⟩ := h'' as' has'
   exact ⟨_, _, hmn, bM, bN, as, has⟩
 #align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_le
+-/
 
+#print Submodule.smithNormalFormOfLE /-
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
@@ -504,7 +533,9 @@ noncomputable def Submodule.smithNormalFormOfLE [Finite ι] (b : Basis ι R M) (
   simp only [snf, Basis.map_apply, Submodule.comapSubtypeEquivOfLe_symm_apply,
     Submodule.coe_smul_of_tower, RelEmbedding.coe_toEmbedding]
 #align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLE
+-/
 
+#print Submodule.smithNormalForm /-
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
@@ -524,11 +555,13 @@ noncomputable def Submodule.smithNormalForm [Finite ι] (b : Basis ι R M) (N :
       Submodule.comapSubtypeEquivOfLe_apply_coe, coe_coe, Basis.reindex_apply,
       Equiv.toEmbedding_apply, Function.Embedding.trans_apply, Equiv.symm_apply_apply]⟩
 #align submodule.smith_normal_form Submodule.smithNormalForm
+-/
 
 section Ideal
 
 variable {S : Type _} [CommRing S] [IsDomain S] [Algebra R S]
 
+#print Ideal.smithNormalForm /-
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -547,9 +580,11 @@ noncomputable def Ideal.smithNormalForm [Fintype ι] (b : Basis ι R S) (I : Ide
   ⟨bS, bI.reindex e, e.symm.toEmbedding.trans f, a ∘ e.symm, fun i => by
     simp only [snf, Basis.coe_reindex, Function.Embedding.trans_apply, Equiv.toEmbedding_apply]⟩
 #align ideal.smith_normal_form Ideal.smithNormalForm
+-/
 
 variable [Finite ι]
 
+#print Ideal.exists_smith_normal_form /-
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -575,7 +610,9 @@ theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I
         simp only [snf, fe, Basis.map_apply, LinearEquiv.restrictScalars_apply,
           Submodule.restrictScalarsEquiv_apply, Basis.coe_reindex]⟩
 #align ideal.exists_smith_normal_form Ideal.exists_smith_normal_form
+-/
 
+#print Ideal.ringBasis /-
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -587,7 +624,9 @@ and `ideal.self_basis_def` for the proof that the inclusion map forms a square d
 noncomputable def Ideal.ringBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) : Basis ι R S :=
   (Ideal.exists_smith_normal_form b I hI).some
 #align ideal.ring_basis Ideal.ringBasis
+-/
 
+#print Ideal.selfBasis /-
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -599,7 +638,9 @@ and `ideal.self_basis_def` for the proof that the inclusion map forms a square d
 noncomputable def Ideal.selfBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) : Basis ι R I :=
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose_spec.some
 #align ideal.self_basis Ideal.selfBasis
+-/
 
+#print Ideal.smithCoeffs /-
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -611,7 +652,9 @@ and `ideal.self_basis_def` for the proof that the inclusion map forms a square d
 noncomputable def Ideal.smithCoeffs (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) : ι → R :=
   (Ideal.exists_smith_normal_form b I hI).choose_spec.some
 #align ideal.smith_coeffs Ideal.smithCoeffs
+-/
 
+#print Ideal.selfBasis_def /-
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -622,7 +665,9 @@ theorem Ideal.selfBasis_def (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
     ∀ i, (Ideal.selfBasis b I hI i : S) = Ideal.smithCoeffs b I hI i • Ideal.ringBasis b I hI i :=
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose_spec.choose_spec
 #align ideal.self_basis_def Ideal.selfBasis_def
+-/
 
+#print Ideal.smithCoeffs_ne_zero /-
 @[simp]
 theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) (i) :
     Ideal.smithCoeffs b I hI i ≠ 0 := by
@@ -631,6 +676,7 @@ theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   refine' Subtype.coe_injective _
   simp [hi]
 #align ideal.smith_coeffs_ne_zero Ideal.smithCoeffs_ne_zero
+-/
 
 instance (F : Type u) [CommRing F] [Algebra F R] (b : Basis ι R S) {I : Ideal S} (hI : I ≠ ⊥) (i) :
     Module F (R ⧸ Ideal.span ({I.smithCoeffs b hI i} : Set R)) := by infer_instance
@@ -642,6 +688,7 @@ end SmithNormal
 
 end PrincipalIdealDomain
 
+#print LinearIndependent.restrict_scalars_algebras /-
 /-- A set of linearly independent vectors in a module `M` over a semiring `S` is also linearly
 independent over a subring `R` of `K`. -/
 theorem LinearIndependent.restrict_scalars_algebras {R S M ι : Type _} [CommSemiring R] [Semiring S]
@@ -650,4 +697,5 @@ theorem LinearIndependent.restrict_scalars_algebras {R S M ι : Type _} [CommSem
     LinearIndependent R v :=
   LinearIndependent.restrict_scalars (by rwa [Algebra.algebraMap_eq_smul_one'] at hinj ) li
 #align linear_independent.restrict_scalars_algebras LinearIndependent.restrict_scalars_algebras
+-/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -236,8 +236,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
         a • ϕ ⟨y', y'M⟩ = ϕ ⟨a • y', _⟩ := (ϕ.map_smul a ⟨y', y'M⟩).symm
         _ = ϕ ⟨y, N_le_M yN⟩ := by simp only [a_smul_y']
         _ = a := ϕy_eq
-        _ = a * 1 := (mul_one a).symm
-        )
+        _ = a * 1 := (mul_one a).symm)
   have ϕy'_ne_zero : ϕ ⟨y', y'M⟩ ≠ 0 := by simpa only [ϕy'_eq] using one_ne_zero
   -- `M' := ker (ϕ : M → R)` is smaller than `M` and `N' := ker (ϕ : N → R)` is smaller than `N`.
   let M' : Submodule R O := ϕ.ker.map M.subtype
@@ -413,7 +412,6 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M
       (∏ j, a j) • s i = (∏ j in {i}ᶜ, a j) • a i • s i := by
         rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]
       _ ∈ N := N.smul_mem _ (ha' i)
-      
   -- Since a submodule of a free `R`-module is free, we get that `A • M` is free
   obtain ⟨n, b : Basis (Fin n) R φ.range⟩ := Submodule.basisOfPidOfLE this sI_basis
   -- hence `M` is free.

mathlib commit https://github.com/leanprover-community/mathlib/commit/31c24aa72e7b3e5ed97a8412470e904f82b81004

@@ -364,7 +364,7 @@ noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι
 
 variable {M}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:638:2: warning: expanding binder collection (i «expr ∉ » I) -/
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
     (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σ n : ℕ, Basis (Fin n) R M := by

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -369,55 +369,55 @@ variable {M}
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
     (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σ n : ℕ, Basis (Fin n) R M := by
   classical
-    -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
-    have := exists_maximal_independent R s
-    let I : Set ι := this.some
-    obtain
-      ⟨indepI : LinearIndependent R (s ∘ coe : I → M), hI :
-        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I)⟩ :=
-      this.some_spec
-    let N := span R (range <| (s ∘ coe : I → M))
-    -- same as `span R (s '' I)` but more convenient
-    let sI : I → N := fun i => ⟨s i.1, subset_span (mem_range_self i)⟩
-    -- `s` restricted to `I`
-    let sI_basis : Basis I R N
-    -- `s` restricted to `I` is a basis of `N`
-    exact Basis.span indepI
-    -- Our first goal is to build `A ≠ 0` such that `A • M ⊆ N`
-    have exists_a : ∀ i : ι, ∃ a : R, a ≠ 0 ∧ a • s i ∈ N :=
-      by
-      intro i
-      by_cases hi : i ∈ I
-      · use 1, zero_ne_one.symm
-        rw [one_smul]
-        exact subset_span (mem_range_self (⟨i, hi⟩ : I))
-      · simpa [image_eq_range s I] using hI i hi
-    choose a ha ha' using exists_a
-    let A := ∏ i, a i
-    have hA : A ≠ 0 := by
-      rw [Finset.prod_ne_zero_iff]
-      simpa using ha
-    -- `M ≃ A • M` because `M` is torsion free and `A ≠ 0`
-    let φ : M →ₗ[R] M := LinearMap.lsmul R M A
-    have : φ.ker = ⊥ := LinearMap.ker_lsmul hA
-    let ψ : M ≃ₗ[R] φ.range := LinearEquiv.ofInjective φ (linear_map.ker_eq_bot.mp this)
-    have : φ.range ≤ N :=
+  -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
+  have := exists_maximal_independent R s
+  let I : Set ι := this.some
+  obtain
+    ⟨indepI : LinearIndependent R (s ∘ coe : I → M), hI :
+      ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I)⟩ :=
+    this.some_spec
+  let N := span R (range <| (s ∘ coe : I → M))
+  -- same as `span R (s '' I)` but more convenient
+  let sI : I → N := fun i => ⟨s i.1, subset_span (mem_range_self i)⟩
+  -- `s` restricted to `I`
+  let sI_basis : Basis I R N
+  -- `s` restricted to `I` is a basis of `N`
+  exact Basis.span indepI
+  -- Our first goal is to build `A ≠ 0` such that `A • M ⊆ N`
+  have exists_a : ∀ i : ι, ∃ a : R, a ≠ 0 ∧ a • s i ∈ N :=
+    by
+    intro i
+    by_cases hi : i ∈ I
+    · use 1, zero_ne_one.symm
+      rw [one_smul]
+      exact subset_span (mem_range_self (⟨i, hi⟩ : I))
+    · simpa [image_eq_range s I] using hI i hi
+  choose a ha ha' using exists_a
+  let A := ∏ i, a i
+  have hA : A ≠ 0 := by
+    rw [Finset.prod_ne_zero_iff]
+    simpa using ha
+  -- `M ≃ A • M` because `M` is torsion free and `A ≠ 0`
+  let φ : M →ₗ[R] M := LinearMap.lsmul R M A
+  have : φ.ker = ⊥ := LinearMap.ker_lsmul hA
+  let ψ : M ≃ₗ[R] φ.range := LinearEquiv.ofInjective φ (linear_map.ker_eq_bot.mp this)
+  have : φ.range ≤ N :=
+    by
+    -- as announced, `A • M ⊆ N`
+    suffices ∀ i, φ (s i) ∈ N
       by
-      -- as announced, `A • M ⊆ N`
-      suffices ∀ i, φ (s i) ∈ N
-        by
-        rw [LinearMap.range_eq_map, ← hs, φ.map_span_le]
-        rintro _ ⟨i, rfl⟩; apply this
-      intro i
-      calc
-        (∏ j, a j) • s i = (∏ j in {i}ᶜ, a j) • a i • s i := by
-          rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]
-        _ ∈ N := N.smul_mem _ (ha' i)
-        
-    -- Since a submodule of a free `R`-module is free, we get that `A • M` is free
-    obtain ⟨n, b : Basis (Fin n) R φ.range⟩ := Submodule.basisOfPidOfLE this sI_basis
-    -- hence `M` is free.
-    exact ⟨n, b.map ψ.symm⟩
+      rw [LinearMap.range_eq_map, ← hs, φ.map_span_le]
+      rintro _ ⟨i, rfl⟩; apply this
+    intro i
+    calc
+      (∏ j, a j) • s i = (∏ j in {i}ᶜ, a j) • a i • s i := by
+        rw [Fintype.prod_eq_prod_compl_mul i, mul_smul]
+      _ ∈ N := N.smul_mem _ (ha' i)
+      
+  -- Since a submodule of a free `R`-module is free, we get that `A • M` is free
+  obtain ⟨n, b : Basis (Fin n) R φ.range⟩ := Submodule.basisOfPidOfLE this sI_basis
+  -- hence `M` is free.
+  exact ⟨n, b.map ψ.symm⟩
 #align module.basis_of_finite_type_torsion_free Module.basisOfFiniteTypeTorsionFree
 
 theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs : span R (range s) = ⊤)

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -74,7 +74,7 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
   rw [Submodule.eq_bot_iff]
   intro x hx
   refine' b.ext_elem fun i => _
-  rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
+  rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ 
   rw [LinearEquiv.map_zero, Finsupp.zero_apply]
   exact
     (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _
@@ -88,7 +88,7 @@ theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M}
   rw [Submodule.eq_bot_iff]
   intro x hx
   refine' congr_arg coe (show (⟨x, hNO hx⟩ : O) = 0 from b.ext_elem fun i => _)
-  rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
+  rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ 
   rw [LinearEquiv.map_zero, Finsupp.zero_apply]
   refine' (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _ _
   exact (LinearMap.mem_submoduleImage_of_le hNO).mpr ⟨x, hx, rfl⟩
@@ -176,16 +176,15 @@ but must also feed in a basis for `M` using `basis_of_pid` to keep the induction
 theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Module R O]
     (M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
     ∃ y ∈ M,
-      ∃ (a : R)(hay : a • y ∈ N),
+      ∃ (a : R) (hay : a • y ∈ N),
         ∃ M' ≤ M,
           ∃ N' ≤ N,
-            ∃ (N'_le_M' : N' ≤ M')(y_ortho_M' :
-              ∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0)(ay_ortho_N' :
-              ∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0),
+            ∃ (N'_le_M' : N' ≤ M') (y_ortho_M' : ∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0)
+              (ay_ortho_N' : ∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0),
               ∀ (n') (bN' : Basis (Fin n') R N'),
                 ∃ bN : Basis (Fin (n' + 1)) R N,
                   ∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
-                    ∃ (hnm : n' + 1 ≤ m' + 1)(bM : Basis (Fin (m' + 1)) R M),
+                    ∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
                       ∀ (as : Fin n' → R)
                         (h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)),
                         ∃ as' : Fin (n' + 1) → R,
@@ -246,7 +245,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   have M'_le_M : M' ≤ M := M.map_subtype_le ϕ.ker
   have N'_le_M' : N' ≤ M' := by
     intro x hx
-    simp only [mem_map, LinearMap.mem_ker] at hx⊢
+    simp only [mem_map, LinearMap.mem_ker] at hx ⊢
     obtain ⟨⟨x, xN⟩, hx, rfl⟩ := hx
     exact ⟨⟨x, N_le_M xN⟩, hx, rfl⟩
   have N'_le_N : N' ≤ N := N.map_subtype_le (ϕ.comp (of_le N_le_M)).ker
@@ -257,7 +256,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
     by
     intro c x xM' hc
     obtain ⟨⟨x, xM⟩, hx', rfl⟩ := submodule.mem_map.mp xM'
-    rw [LinearMap.mem_ker] at hx'
+    rw [LinearMap.mem_ker] at hx' 
     have hc' : (c • ⟨y', y'M⟩ + ⟨x, xM⟩ : M) = 0 := Subtype.coe_injective hc
     simpa only [LinearMap.map_add, LinearMap.map_zero, LinearMap.map_smul, smul_eq_mul, add_zero,
       mul_eq_zero, ϕy'_ne_zero, hx', or_false_iff] using congr_arg ϕ hc'
@@ -272,7 +271,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   · refine' Basis.mkFinConsOfLe y yN bN' N'_le_N _ _
     · intro c z zN' hc
       refine' ay'_ortho_N' c z zN' _
-      rwa [← a_smul_y'] at hc
+      rwa [← a_smul_y'] at hc 
     · intro z zN
       obtain ⟨b, hb⟩ : _ ∣ ϕ ⟨z, N_le_M zN⟩ := generator_submodule_image_dvd_of_mem N_le_M ϕ zN
       refine' ⟨-b, submodule.mem_map.mpr ⟨⟨_, N.sub_mem zN (N.smul_mem b yN)⟩, _, _⟩⟩
@@ -330,7 +329,7 @@ if `R` is a principal ideal domain.
 See also the stronger version `submodule.smith_normal_form`.
 -/
 noncomputable def Submodule.basisOfPid {ι : Type _} [Finite ι] (b : Basis ι R M)
-    (N : Submodule R M) : Σn : ℕ, Basis (Fin n) R N :=
+    (N : Submodule R M) : Σ n : ℕ, Basis (Fin n) R N :=
   ⟨_, (N.nonempty_basis_of_pid b).choose_spec.some⟩
 #align submodule.basis_of_pid Submodule.basisOfPid
 -/
@@ -350,7 +349,7 @@ if `R` is a principal ideal domain.
 See also the stronger version `submodule.smith_normal_form_of_le`.
 -/
 noncomputable def Submodule.basisOfPidOfLE {ι : Type _} [Finite ι] {N O : Submodule R M}
-    (hNO : N ≤ O) (b : Basis ι R O) : Σn : ℕ, Basis (Fin n) R N :=
+    (hNO : N ≤ O) (b : Basis ι R O) : Σ n : ℕ, Basis (Fin n) R N :=
   let ⟨n, bN'⟩ := Submodule.basisOfPid b (N.comap O.Subtype)
   ⟨n, bN'.map (Submodule.comapSubtypeEquivOfLe hNO)⟩
 #align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLE
@@ -359,7 +358,7 @@ noncomputable def Submodule.basisOfPidOfLE {ι : Type _} [Finite ι] {N O : Subm
 if `R` is a principal ideal domain. -/
 noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι → M}
     (hb : LinearIndependent R b) {N : Submodule R M} (le : N ≤ Submodule.span R (Set.range b)) :
-    Σn : ℕ, Basis (Fin n) R N :=
+    Σ n : ℕ, Basis (Fin n) R N :=
   Submodule.basisOfPidOfLE le (Basis.span hb)
 #align submodule.basis_of_pid_of_le_span Submodule.basisOfPidOfLESpan
 
@@ -368,7 +367,7 @@ variable {M}
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
-    (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M := by
+    (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σ n : ℕ, Basis (Fin n) R M := by
   classical
     -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
     have := exists_maximal_independent R s
@@ -425,20 +424,20 @@ theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs :
     [NoZeroSMulDivisors R M] : Module.Free R M :=
   by
   cases nonempty_fintype ι
-  obtain ⟨n, b⟩ : Σn, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree hs
+  obtain ⟨n, b⟩ : Σ n, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree hs
   exact Module.Free.of_basis b
 #align module.free_of_finite_type_torsion_free Module.free_of_finite_type_torsion_free
 
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree' [Module.Finite R M]
-    [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M :=
+    [NoZeroSMulDivisors R M] : Σ n : ℕ, Basis (Fin n) R M :=
   Module.basisOfFiniteTypeTorsionFree Module.Finite.exists_fin.choose_spec.choose_spec
 #align module.basis_of_finite_type_torsion_free' Module.basisOfFiniteTypeTorsionFree'
 
 theorem Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMulDivisors R M] :
     Module.Free R M :=
   by
-  obtain ⟨n, b⟩ : Σn, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree'
+  obtain ⟨n, b⟩ : Σ n, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree'
   exact Module.Free.of_basis b
 #align module.free_of_finite_type_torsion_free' Module.free_of_finite_type_torsion_free'
 
@@ -467,7 +466,7 @@ This is a strengthening of `submodule.basis_of_pid_of_le`.
 -/
 theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M) (N O : Submodule R M)
     (N_le_O : N ≤ O) :
-    ∃ (n o : ℕ)(hno : n ≤ o)(bO : Basis (Fin o) R O)(bN : Basis (Fin n) R N)(a : Fin n → R),
+    ∃ (n o : ℕ) (hno : n ≤ o) (bO : Basis (Fin o) R O) (bN : Basis (Fin n) R N) (a : Fin n → R),
       ∀ i, (bN i : M) = a i • bO (Fin.castLE hno i) :=
   by
   cases nonempty_fintype ι
@@ -497,7 +496,7 @@ need to map `N` into a submodule of `O`.
 This is a strengthening of `submodule.basis_of_pid_of_le`.
 -/
 noncomputable def Submodule.smithNormalFormOfLE [Finite ι] (b : Basis ι R M) (N O : Submodule R M)
-    (N_le_O : N ≤ O) : Σo n : ℕ, Basis.SmithNormalForm (N.comap O.Subtype) (Fin o) n :=
+    (N_le_O : N ≤ O) : Σ o n : ℕ, Basis.SmithNormalForm (N.comap O.Subtype) (Fin o) n :=
   by
   choose n o hno bO bN a snf using N.exists_smith_normal_form_of_le b O N_le_O
   refine'
@@ -518,7 +517,7 @@ See also `ideal.smith_normal_form`, which moreover proves that the dimension of
 an ideal is the same as the dimension of the whole ring.
 -/
 noncomputable def Submodule.smithNormalForm [Finite ι] (b : Basis ι R M) (N : Submodule R M) :
-    Σn : ℕ, Basis.SmithNormalForm N ι n :=
+    Σ n : ℕ, Basis.SmithNormalForm N ι n :=
   let ⟨m, n, bM, bN, f, a, snf⟩ := N.smithNormalFormOfLE b ⊤ le_top
   let bM' := bM.map (LinearEquiv.ofTop _ rfl)
   let e := bM'.indexEquiv b
@@ -565,7 +564,7 @@ The definitions `ideal.ring_basis`, `ideal.self_basis`, `ideal.smith_coeffs` are
 choices of values for this existential quantifier.
 -/
 theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
-    ∃ (b' : Basis ι R S)(a : ι → R)(ab' : Basis ι R I), ∀ i, (ab' i : S) = a i • b' i := by
+    ∃ (b' : Basis ι R S) (a : ι → R) (ab' : Basis ι R I), ∀ i, (ab' i : S) = a i • b' i := by
   cases nonempty_fintype ι <;>
     exact
       let ⟨bS, bI, f, a, snf⟩ := I.smith_normal_form b hI
@@ -651,6 +650,6 @@ theorem LinearIndependent.restrict_scalars_algebras {R S M ι : Type _} [CommSem
     [AddCommMonoid M] [Algebra R S] [Module R M] [Module S M] [IsScalarTower R S M]
     (hinj : Function.Injective (algebraMap R S)) {v : ι → M} (li : LinearIndependent S v) :
     LinearIndependent R v :=
-  LinearIndependent.restrict_scalars (by rwa [Algebra.algebraMap_eq_smul_one'] at hinj) li
+  LinearIndependent.restrict_scalars (by rwa [Algebra.algebraMap_eq_smul_one'] at hinj ) li
 #align linear_independent.restrict_scalars_algebras LinearIndependent.restrict_scalars_algebras
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/88a563b158f59f2983cfad685664da95502e8cdd

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 
 ! This file was ported from Lean 3 source module linear_algebra.free_module.pid
-! leanprover-community/mathlib commit 86d1873c01a723aba6788f0b9051ae3d23b4c1c3
+! leanprover-community/mathlib commit d87199d51218d36a0a42c66c82d147b5a7ff87b3
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -635,6 +635,10 @@ theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   simp [hi]
 #align ideal.smith_coeffs_ne_zero Ideal.smithCoeffs_ne_zero
 
+instance (F : Type u) [CommRing F] [Algebra F R] (b : Basis ι R S) {I : Ideal S} (hI : I ≠ ⊥) (i) :
+    Module F (R ⧸ Ideal.span ({I.smithCoeffs b hI i} : Set R)) := by infer_instance
+
+-- quotient.module' _
 end Ideal
 
 end SmithNormal

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -55,7 +55,7 @@ free module, finitely generated module, rank, structure theorem
 -/
 
 
-open BigOperators
+open scoped BigOperators
 
 universe u v
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -67,9 +67,6 @@ variable {ι : Type _} (b : Basis ι R M)
 
 open Submodule.IsPrincipal Submodule
 
-/- warning: eq_bot_of_generator_maximal_map_eq_zero -> eq_bot_of_generator_maximal_map_eq_zero is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
     {ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
     (hgen : generator (N.map ϕ) = (0 : R)) : N = ⊥ :=
@@ -84,9 +81,6 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
       ⟨x, hx, rfl⟩
 #align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zero
 
-/- warning: eq_bot_of_generator_maximal_submodule_image_eq_zero -> eq_bot_of_generator_maximal_submoduleImage_eq_zero is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_submodule_image_eq_zero eq_bot_of_generator_maximal_submoduleImage_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
     (hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (hgen : generator (ϕ.submoduleImage N) = 0) : N = ⊥ :=
@@ -132,9 +126,6 @@ variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal
 
-/- warning: generator_maximal_submodule_image_dvd -> generator_maximal_submoduleImage_dvd is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvdₓ'. -/
 theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
     (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (y : M) (yN : y ∈ N)
@@ -171,9 +162,6 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
   · exact subset_span (mem_insert _ _)
 #align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvd
 
-/- warning: submodule.basis_of_pid_aux -> Submodule.basis_of_pid_aux is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_aux Submodule.basis_of_pid_auxₓ'. -/
 /-- The induction hypothesis of `submodule.basis_of_pid` and `submodule.smith_normal_form`.
 
 Basically, it says: let `N ≤ M` be a pair of submodules, then we can find a pair of
@@ -310,9 +298,6 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   · rw [Fin.castLE_succ]; simp only [Fin.cons_succ, coe_of_le, h i]
 #align submodule.basis_of_pid_aux Submodule.basis_of_pid_aux
 
-/- warning: submodule.nonempty_basis_of_pid -> Submodule.nonempty_basis_of_pid is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.nonempty_basis_of_pid Submodule.nonempty_basis_of_pidₓ'. -/
 /-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
 
@@ -350,9 +335,6 @@ noncomputable def Submodule.basisOfPid {ι : Type _} [Finite ι] (b : Basis ι R
 #align submodule.basis_of_pid Submodule.basisOfPid
 -/
 
-/- warning: submodule.basis_of_pid_bot -> Submodule.basisOfPid_bot is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_bot Submodule.basisOfPid_botₓ'. -/
 theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
     Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ :=
   by
@@ -362,9 +344,6 @@ theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
   exact Sigma.eq rfl (Basis.eq_of_apply_eq <| finZeroElim)
 #align submodule.basis_of_pid_bot Submodule.basisOfPid_bot
 
-/- warning: submodule.basis_of_pid_of_le -> Submodule.basisOfPidOfLE is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLEₓ'. -/
 /-- A submodule inside a free `R`-submodule of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
 
@@ -376,9 +355,6 @@ noncomputable def Submodule.basisOfPidOfLE {ι : Type _} [Finite ι] {N O : Subm
   ⟨n, bN'.map (Submodule.comapSubtypeEquivOfLe hNO)⟩
 #align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLE
 
-/- warning: submodule.basis_of_pid_of_le_span -> Submodule.basisOfPidOfLESpan is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_of_le_span Submodule.basisOfPidOfLESpanₓ'. -/
 /-- A submodule inside the span of a linear independent family is a free `R`-module of finite rank,
 if `R` is a principal ideal domain. -/
 noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι → M}
@@ -389,12 +365,6 @@ noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι
 
 variable {M}
 
-/- warning: module.basis_of_finite_type_torsion_free -> Module.basisOfFiniteTypeTorsionFree is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Fintype.{u1} ι] {s : ι -> M}, (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.span.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 (Set.range.{u3, succ u1} M ι s)) (Top.top.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.hasTop.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (SubNegMonoid.toAddMonoid.{u3} M (AddGroup.toSubNegMonoid.{u3} M (AddCommGroup.toAddGroup.{u3} M _inst_4))))) (SMulZeroClass.toHasSmul.{u2, u3} R M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (Module.toMulActionWithZero.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))], Sigma.{0, max u2 u3} Nat (fun (n : Nat) => Basis.{0, u2, u3} (Fin n) R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))
-but is expected to have type
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Fintype.{u1} ι] {s : ι -> M}, (Eq.{succ u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.span.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 (Set.range.{u3, succ u1} M ι s)) (Top.top.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.instTopSubmodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (SMulZeroClass.toSMul.{u2, u3} R M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (Module.toMulActionWithZero.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))], Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.{0, u2, u3} (Fin n) R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))
-Case conversion may be inaccurate. Consider using '#align module.basis_of_finite_type_torsion_free Module.basisOfFiniteTypeTorsionFreeₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
@@ -451,12 +421,6 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M
     exact ⟨n, b.map ψ.symm⟩
 #align module.basis_of_finite_type_torsion_free Module.basisOfFiniteTypeTorsionFree
 
-/- warning: module.free_of_finite_type_torsion_free -> Module.free_of_finite_type_torsion_free is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι] {s : ι -> M}, (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.span.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 (Set.range.{u3, succ u1} M ι s)) (Top.top.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.hasTop.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (SubNegMonoid.toAddMonoid.{u3} M (AddGroup.toSubNegMonoid.{u3} M (AddCommGroup.toAddGroup.{u3} M _inst_4))))) (SMulZeroClass.toHasSmul.{u2, u3} R M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (Module.toMulActionWithZero.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))], Module.Free.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)
-but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Finite.{succ u3} ι] {s : ι -> M}, (Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.span.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι s)) (Top.top.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.instTopSubmodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Module.Free.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
-Case conversion may be inaccurate. Consider using '#align module.free_of_finite_type_torsion_free Module.free_of_finite_type_torsion_freeₓ'. -/
 theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs : span R (range s) = ⊤)
     [NoZeroSMulDivisors R M] : Module.Free R M :=
   by
@@ -465,24 +429,12 @@ theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs :
   exact Module.Free.of_basis b
 #align module.free_of_finite_type_torsion_free Module.free_of_finite_type_torsion_free
 
-/- warning: module.basis_of_finite_type_torsion_free' -> Module.basisOfFiniteTypeTorsionFree' is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Module.Finite.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4))))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
-but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Module.Finite.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
-Case conversion may be inaccurate. Consider using '#align module.basis_of_finite_type_torsion_free' Module.basisOfFiniteTypeTorsionFree'ₓ'. -/
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree' [Module.Finite R M]
     [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M :=
   Module.basisOfFiniteTypeTorsionFree Module.Finite.exists_fin.choose_spec.choose_spec
 #align module.basis_of_finite_type_torsion_free' Module.basisOfFiniteTypeTorsionFree'
 
-/- warning: module.free_of_finite_type_torsion_free' -> Module.free_of_finite_type_torsion_free' is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align module.free_of_finite_type_torsion_free' Module.free_of_finite_type_torsion_free'ₓ'. -/
 theorem Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMulDivisors R M] :
     Module.Free R M :=
   by
@@ -492,12 +444,6 @@ theorem Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMul
 
 section SmithNormal
 
-/- warning: basis.smith_normal_form -> Basis.SmithNormalForm is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align basis.smith_normal_form Basis.SmithNormalFormₓ'. -/
 /-- A Smith normal form basis for a submodule `N` of a module `M` consists of
 bases for `M` and `N` such that the inclusion map `N → M` can be written as a
 (rectangular) matrix with `a` along the diagonal: in Smith normal form. -/
@@ -510,9 +456,6 @@ structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type _) (n : ℕ) wher
   snf : ∀ i, (bN i : M) = a i • bM (f i)
 #align basis.smith_normal_form Basis.SmithNormalForm
 
-/- warning: submodule.exists_smith_normal_form_of_le -> Submodule.exists_smith_normal_form_of_le is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_leₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
@@ -544,9 +487,6 @@ theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M)
   exact ⟨_, _, hmn, bM, bN, as, has⟩
 #align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_le
 
-/- warning: submodule.smith_normal_form_of_le -> Submodule.smithNormalFormOfLE is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLEₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
@@ -568,12 +508,6 @@ noncomputable def Submodule.smithNormalFormOfLE [Finite ι] (b : Basis ι R M) (
     Submodule.coe_smul_of_tower, RelEmbedding.coe_toEmbedding]
 #align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLE
 
-/- warning: submodule.smith_normal_form -> Submodule.smithNormalForm is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align submodule.smith_normal_form Submodule.smithNormalFormₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
@@ -598,9 +532,6 @@ section Ideal
 
 variable {S : Type _} [CommRing S] [IsDomain S] [Algebra R S]
 
-/- warning: ideal.smith_normal_form -> Ideal.smithNormalForm is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align ideal.smith_normal_form Ideal.smithNormalFormₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -622,9 +553,6 @@ noncomputable def Ideal.smithNormalForm [Fintype ι] (b : Basis ι R S) (I : Ide
 
 variable [Finite ι]
 
-/- warning: ideal.exists_smith_normal_form -> Ideal.exists_smith_normal_form is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align ideal.exists_smith_normal_form Ideal.exists_smith_normal_formₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -651,12 +579,6 @@ theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I
           Submodule.restrictScalarsEquiv_apply, Basis.coe_reindex]⟩
 #align ideal.exists_smith_normal_form Ideal.exists_smith_normal_form
 
-/- warning: ideal.ring_basis -> Ideal.ringBasis is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align ideal.ring_basis Ideal.ringBasisₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -669,9 +591,6 @@ noncomputable def Ideal.ringBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   (Ideal.exists_smith_normal_form b I hI).some
 #align ideal.ring_basis Ideal.ringBasis
 
-/- warning: ideal.self_basis -> Ideal.selfBasis is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align ideal.self_basis Ideal.selfBasisₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -684,12 +603,6 @@ noncomputable def Ideal.selfBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose_spec.some
 #align ideal.self_basis Ideal.selfBasis
 
-/- warning: ideal.smith_coeffs -> Ideal.smithCoeffs is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align ideal.smith_coeffs Ideal.smithCoeffsₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -702,9 +615,6 @@ noncomputable def Ideal.smithCoeffs (b : Basis ι R S) (I : Ideal S) (hI : I ≠
   (Ideal.exists_smith_normal_form b I hI).choose_spec.some
 #align ideal.smith_coeffs Ideal.smithCoeffs
 
-/- warning: ideal.self_basis_def -> Ideal.selfBasis_def is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align ideal.self_basis_def Ideal.selfBasis_defₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -716,12 +626,6 @@ theorem Ideal.selfBasis_def (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose_spec.choose_spec
 #align ideal.self_basis_def Ideal.selfBasis_def
 
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 @[simp]
 theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) (i) :
     Ideal.smithCoeffs b I hI i ≠ 0 := by
@@ -737,12 +641,6 @@ end SmithNormal
 
 end PrincipalIdealDomain
 
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 /-- A set of linearly independent vectors in a module `M` over a semiring `S` is also linearly
 independent over a subring `R` of `K`. -/
 theorem LinearIndependent.restrict_scalars_algebras {R S M ι : Type _} [CommSemiring R] [Semiring S]

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -234,18 +234,13 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   let y' : O := ∑ i, c i • b'M i
   have y'M : y' ∈ M := M.sum_mem fun i _ => M.smul_mem (c i) (b'M i).2
   have mk_y' : (⟨y', y'M⟩ : M) = ∑ i, c i • b'M i :=
-    Subtype.ext
-      (show y' = M.subtype _
-        by
-        simp only [LinearMap.map_sum, LinearMap.map_smul]
-        rfl)
+    Subtype.ext (show y' = M.subtype _ by simp only [LinearMap.map_sum, LinearMap.map_smul]; rfl)
   have a_smul_y' : a • y' = y :=
     by
     refine' congr_arg coe (show (a • ⟨y', y'M⟩ : M) = ⟨y, N_le_M yN⟩ from _)
     rw [← b'M.sum_repr ⟨y, N_le_M yN⟩, mk_y', Finset.smul_sum]
     refine' Finset.sum_congr rfl fun i _ => _
-    rw [← mul_smul, ← hc]
-    rfl
+    rw [← mul_smul, ← hc]; rfl
   -- We found an `y` and an `a`!
   refine' ⟨y', y'M, a, a_smul_y'.symm ▸ yN, _⟩
   have ϕy'_eq : ϕ ⟨y', y'M⟩ = 1 :=
@@ -295,8 +290,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
       refine' ⟨-b, submodule.mem_map.mpr ⟨⟨_, N.sub_mem zN (N.smul_mem b yN)⟩, _, _⟩⟩
       · refine' linear_map.mem_ker.mpr (show ϕ (⟨z, N_le_M zN⟩ - b • ⟨y, N_le_M yN⟩) = 0 from _)
         rw [LinearMap.map_sub, LinearMap.map_smul, hb, ϕy_eq, smul_eq_mul, mul_comm, sub_self]
-      · simp only [sub_eq_add_neg, neg_smul]
-        rfl
+      · simp only [sub_eq_add_neg, neg_smul]; rfl
   -- And extend a basis for `M'` with `y'`
   intro m' hn'm' bM'
   refine' ⟨Nat.succ_le_succ hn'm', _, _⟩
@@ -304,8 +298,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
     intro z zM
     refine' ⟨-ϕ ⟨z, zM⟩, ⟨⟨z, zM⟩ - ϕ ⟨z, zM⟩ • ⟨y', y'M⟩, linear_map.mem_ker.mpr _, _⟩⟩
     · rw [LinearMap.map_sub, LinearMap.map_smul, ϕy'_eq, smul_eq_mul, mul_one, sub_self]
-    · rw [LinearMap.map_sub, LinearMap.map_smul, sub_eq_add_neg, neg_smul]
-      rfl
+    · rw [LinearMap.map_sub, LinearMap.map_smul, sub_eq_add_neg, neg_smul]; rfl
   -- It remains to show the extended bases are compatible with each other.
   intro as h
   refine' ⟨Fin.cons a as, _⟩
@@ -314,8 +307,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   refine' Fin.cases _ (fun i => _) i
   · simp only [Fin.cons_zero, Fin.castLE_zero]
     exact a_smul_y'.symm
-  · rw [Fin.castLE_succ]
-    simp only [Fin.cons_succ, coe_of_le, h i]
+  · rw [Fin.castLE_succ]; simp only [Fin.cons_succ, coe_of_le, h i]
 #align submodule.basis_of_pid_aux Submodule.basis_of_pid_aux
 
 /- warning: submodule.nonempty_basis_of_pid -> Submodule.nonempty_basis_of_pid is a dubious translation:
@@ -338,8 +330,7 @@ theorem Submodule.nonempty_basis_of_pid {ι : Type _} [Finite ι] (b : Basis ι
   intro N ih
   let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
   by_cases N_bot : N = ⊥
-  · subst N_bot
-    exact ⟨0, ⟨Basis.empty _⟩⟩
+  · subst N_bot; exact ⟨0, ⟨Basis.empty _⟩⟩
   obtain ⟨y, -, a, hay, M', -, N', N'_le_N, -, -, ay_ortho, h'⟩ :=
     Submodule.basis_of_pid_aux ⊤ N b' N_bot le_top
   obtain ⟨n', ⟨bN'⟩⟩ := ih N' N'_le_N _ hay ay_ortho
@@ -447,8 +438,7 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M
       suffices ∀ i, φ (s i) ∈ N
         by
         rw [LinearMap.range_eq_map, ← hs, φ.map_span_le]
-        rintro _ ⟨i, rfl⟩
-        apply this
+        rintro _ ⟨i, rfl⟩; apply this
       intro i
       calc
         (∏ j, a j) • s i = (∏ j in {i}ᶜ, a j) • a i • s i := by

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -68,10 +68,7 @@ variable {ι : Type _} (b : Basis ι R M)
 open Submodule.IsPrincipal Submodule
 
 /- warning: eq_bot_of_generator_maximal_map_eq_zero -> eq_bot_of_generator_maximal_map_eq_zero is a dubious translation:
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(NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.semilinearMapClass.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.semilinearMapClass.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.semilinearMapClass.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} R (Ring.toAddCommGroup.{u2} R _inst_1)) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.semilinearMapClass.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
     {ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
@@ -88,10 +85,7 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
 #align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zero
 
 /- warning: eq_bot_of_generator_maximal_submodule_image_eq_zero -> eq_bot_of_generator_maximal_submoduleImage_eq_zero is a dubious translation:
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(Ring.toSemiring.{u1} R _inst_1)) O ϕ N) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R 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-but is expected to have type
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(NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_submodule_image_eq_zero eq_bot_of_generator_maximal_submoduleImage_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
     (hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
@@ -139,10 +133,7 @@ variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
 open Submodule.IsPrincipal
 
 /- warning: generator_maximal_submodule_image_dvd -> generator_maximal_submoduleImage_dvd is a dubious translation:
-lean 3 declaration is
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(PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N O) {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M 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(Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O) y (hNO y yN))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (CommRing.toRing.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R 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+<too large>
 Case conversion may be inaccurate. Consider using '#align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvdₓ'. -/
 theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
     (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
@@ -181,10 +172,7 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
 #align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvd
 
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(SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y)) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O 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-  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_6 : Finite.{succ u3} ι] {O : Type.{u2}} [_inst_7 : AddCommGroup.{u2} O] [_inst_8 : Module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7)] (M : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (N : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8), (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x M)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M) (Submodule.module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M)) -> (Ne.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) N (Bot.bot.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.instBotSubmodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1538 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_ay_ortho_N' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => forall (n' : Nat) (bN' : Basis.{0, u1, u2} (Fin n') R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R 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 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_aux Submodule.basis_of_pid_auxₓ'. -/
 /-- The induction hypothesis of `submodule.basis_of_pid` and `submodule.smith_normal_form`.
 
@@ -331,10 +319,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
 #align submodule.basis_of_pid_aux Submodule.basis_of_pid_aux
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.nonempty_basis_of_pid Submodule.nonempty_basis_of_pidₓ'. -/
 /-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
@@ -375,10 +360,7 @@ noncomputable def Submodule.basisOfPid {ι : Type _} [Finite ι] (b : Basis ι R
 -/
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_bot Submodule.basisOfPid_botₓ'. -/
 theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
     Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ :=
@@ -390,10 +372,7 @@ theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
 #align submodule.basis_of_pid_bot Submodule.basisOfPid_bot
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLEₓ'. -/
 /-- A submodule inside a free `R`-submodule of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
@@ -407,10 +386,7 @@ noncomputable def Submodule.basisOfPidOfLE {ι : Type _} [Finite ι] {N O : Subm
 #align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLE
 
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 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_of_le_span Submodule.basisOfPidOfLESpanₓ'. -/
 /-- A submodule inside the span of a linear independent family is a free `R`-module of finite rank,
 if `R` is a principal ideal domain. -/
@@ -545,10 +521,7 @@ structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type _) (n : ℕ) wher
 #align basis.smith_normal_form Basis.SmithNormalForm
 
 /- warning: submodule.exists_smith_normal_form_of_le -> Submodule.exists_smith_normal_form_of_le is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_leₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
@@ -582,10 +555,7 @@ theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M)
 #align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_le
 
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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))) N O) -> (Sigma.{0, max u3 u2} Nat (fun (o : Nat) => Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.comap.{u2, u2, u3, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R 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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.semilinearMapClass.{u2, u2, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLEₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
@@ -639,10 +609,7 @@ section Ideal
 variable {S : Type _} [CommRing S] [IsDomain S] [Algebra R S]
 
 /- warning: ideal.smith_normal_form -> Ideal.smithNormalForm is a dubious translation:
-lean 3 declaration is
-  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {S : Type.{u_4}} [_inst_6 : CommRing.{u_4} S] [_inst_7 : IsDomain.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))] [_inst_8 : Algebra.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))] [_inst_9 : Fintype.{u_1} ι], (Basis.{u_1, u_2, u_4} ι R S (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_4} S (NonUnitalNonAssocRing.toAddCommGroup.{u_4} S (NonAssocRing.toNonUnitalNonAssocRing.{u_4} S (Ring.toNonAssocRing.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))), (Ne.{succ u_4} (Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) I (Bot.bot.{u_4} (Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) (Submodule.hasBot.{u_4, u_4} S S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Semiring.toModule.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))) -> (Basis.SmithNormalForm.{u_2, u_4, u_1} R _inst_1 _inst_2 _inst_3 S (NonUnitalNonAssocRing.toAddCommGroup.{u_4} S (NonAssocRing.toNonUnitalNonAssocRing.{u_4} S (Ring.toNonAssocRing.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8) (Submodule.restrictScalars.{u_2, u_4, u_4} R S S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8) (Semiring.toModule.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) (SMulZeroClass.toHasSmul.{u_2, u_4} R S (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u_2, u_4} R S (MulZeroClass.toHasZero.{u_2} R (MulZeroOneClass.toMulZeroClass.{u_2} R (MonoidWithZero.toMulZeroOneClass.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)))))) (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (MulActionWithZero.toSMulWithZero.{u_2, u_4} R S (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1))) (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (Module.toMulActionWithZero.{u_2, u_4} R S (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8))))) (Ideal.smithNormalForm._proof_1.{u_2, u_4} R _inst_1 S _inst_6 _inst_8) I) ι (Fintype.card.{u_1} ι _inst_9)))
-but is expected to have type
-  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {S : Type.{u_3}} [_inst_6 : CommRing.{u_3} S] [_inst_7 : IsDomain.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))] [_inst_8 : Algebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))] [_inst_9 : Fintype.{u_1} ι], (Basis.{u_1, u_2, u_3} ι R S (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonAssocRing.toNonUnitalNonAssocRing.{u_3} S (Ring.toNonAssocRing.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))), (Ne.{succ u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))) I (Bot.bot.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))) (Submodule.instBotSubmodule.{u_3, u_3} S S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))))) (Semiring.toModule.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)))))) -> (Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 S (Ring.toAddCommGroup.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) _inst_8) (Submodule.restrictScalars.{u_2, u_3, u_3} R S S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))))) (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) _inst_8) (Semiring.toModule.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))) (Algebra.toSMul.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) _inst_8) (IsScalarTower.right.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) _inst_8) I) ι (Fintype.card.{u_1} ι _inst_9)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align ideal.smith_normal_form Ideal.smithNormalFormₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -666,10 +633,7 @@ noncomputable def Ideal.smithNormalForm [Fintype ι] (b : Basis ι R S) (I : Ide
 variable [Finite ι]
 
 /- warning: ideal.exists_smith_normal_form -> Ideal.exists_smith_normal_form is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align ideal.exists_smith_normal_form Ideal.exists_smith_normal_formₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -716,10 +680,7 @@ noncomputable def Ideal.ringBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
 #align ideal.ring_basis Ideal.ringBasis
 
 /- warning: ideal.self_basis -> Ideal.selfBasis is a dubious translation:
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 Case conversion may be inaccurate. Consider using '#align ideal.self_basis Ideal.selfBasisₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -752,10 +713,7 @@ noncomputable def Ideal.smithCoeffs (b : Basis ι R S) (I : Ideal S) (hI : I ≠
 #align ideal.smith_coeffs Ideal.smithCoeffs
 
 /- warning: ideal.self_basis_def -> Ideal.selfBasis_def is a dubious translation:
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(CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (fun (_x : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) => ι -> S) (FunLike.hasCoeToFun.{max (succ u1) (succ u2) (succ u3), succ u1, succ u3} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S 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+<too large>
 Case conversion may be inaccurate. Consider using '#align ideal.self_basis_def Ideal.selfBasis_defₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can

mathlib commit https://github.com/leanprover-community/mathlib/commit/8d33f09cd7089ecf074b4791907588245aec5d1b

@@ -142,7 +142,7 @@ open Submodule.IsPrincipal
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N O) {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Preorder.toHasLt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ϕ N) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ψ N))) -> (forall [_inst_6 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ϕ N)] (y : M) (yN : Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) y N), (Eq.{succ u1} R (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M 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(Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O) y (hNO y yN))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (CommRing.toRing.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R 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(NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (CommRing.toRing.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ϕ N) _inst_6) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) 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(Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) => (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ψ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {N : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} {O : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} (hNO : LE.le.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) {ϕ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))}, (forall (ψ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ψ N))) -> (forall [_inst_6 : Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) 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(AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) (fun (_x : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} R R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) ψ (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {N : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} {O : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} (hNO : LE.le.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) {ϕ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))}, (forall (ψ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ψ N))) -> (forall [_inst_6 : Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N)] (y : M) (yN : Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) y N), (Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => R) (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN))) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, 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(CommRing.toRing.{u2} R _inst_1) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N) _inst_6)) -> (forall (ψ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : 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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) ψ (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
 Case conversion may be inaccurate. Consider using '#align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvdₓ'. -/
 theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
     (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -378,7 +378,7 @@ noncomputable def Submodule.basisOfPid {ι : Type _} [Finite ι] (b : Basis ι R
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] (b : Basis.{u3, u1, u2} ι R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5), Eq.{succ (max u1 u2)} (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) 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_inst_5))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) (Submodule.basisOfPid.{u1, u2, u3} R _inst_1 _inst_2 _inst_3 M _inst_4 _inst_5 ι _inst_6 b (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (Sigma.mk.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R 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(AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (Submodule.uniqueBot.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) Fin.isEmpty))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5), Eq.{max (succ u2) (succ u1)} (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u2, u1} (Fin n) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} 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(Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Sigma.mk.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u2, u1} (Fin n) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Basis.empty.{0, u2, u1} (Fin (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Unique.instSubsingleton.{succ u1} (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (Submodule.uniqueBot.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) instIsEmptyFinOfNatNatInstOfNatNat))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5), Eq.{max (succ u2) (succ u1)} (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u2, u1} (Fin n) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) (Submodule.basisOfPid.{u2, u1, u3} R _inst_1 _inst_2 _inst_3 M _inst_4 _inst_5 ι _inst_6 b (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Sigma.mk.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u2, u1} (Fin n) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) 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(CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Basis.empty.{0, u2, u1} (Fin (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Unique.instSubsingleton.{succ u1} (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (Submodule.uniqueBot.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) Fin.isEmpty))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_bot Submodule.basisOfPid_botₓ'. -/
 theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
     Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/95a87616d63b3cb49d3fe678d416fbe9c4217bf4

@@ -184,7 +184,7 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] [_inst_6 : Finite.{succ u1} ι] {O : Type.{u3}} [_inst_7 : AddCommGroup.{u3} O] [_inst_8 : Module.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)] (M : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (N : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8), (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) M) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8 M) (Submodule.module.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8 M)) -> (Ne.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) N (Bot.bot.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Submodule.hasBot.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8))) -> (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N M) -> (Exists.{succ u3} O (fun (y : O) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) (fun (H : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) => Exists.{succ u2} R (fun (a : R) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) (fun (hay : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (M' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (N' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') (fun (N'_le_M' : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) (fun (y_ortho_M' : forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z N') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y)) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) (fun (ay_ortho_N' : forall (c : R) (z : O), (Membership.Mem.{u3, u3} O 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(One.one.{0} Nat Nat.hasOne))))))) => (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) hnm) i)))))))))))))))))))))
 but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_6 : Finite.{succ u3} ι] {O : Type.{u2}} [_inst_7 : AddCommGroup.{u2} O] [_inst_8 : Module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7)] (M : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (N : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8), (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x M)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M) (Submodule.module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M)) -> (Ne.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) N (Bot.bot.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.instBotSubmodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1538 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_ay_ortho_N' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} 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0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
+  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_6 : Finite.{succ u3} ι] {O : Type.{u2}} [_inst_7 : AddCommGroup.{u2} O] [_inst_8 : Module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7)] (M : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (N : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8), (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x M)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M) (Submodule.module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M)) -> (Ne.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) N (Bot.bot.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.instBotSubmodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1538 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_ay_ortho_N' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} 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instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_aux Submodule.basis_of_pid_auxₓ'. -/
 /-- The induction hypothesis of `submodule.basis_of_pid` and `submodule.smith_normal_form`.
 
@@ -548,7 +548,7 @@ structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type _) (n : ℕ) wher
 lean 3 declaration is
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 but is expected to have type
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i))))))))))))
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] [_inst_6 : Finite.{succ u3} ι], (Basis.{u3, u2, u1} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (O : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5), (LE.le.{u1} (Submodule.{u2, u1} R M 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_inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) -> (Exists.{1} Nat (fun (n : Nat) => Exists.{1} Nat (fun (o : Nat) => Exists.{0} (LE.le.{0} Nat instLENat n o) (fun (hno : LE.le.{0} Nat instLENat n o) => Exists.{max (succ u2) (succ u1)} (Basis.{0, u2, u1} (Fin o) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M 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i))))))))))))
 Case conversion may be inaccurate. Consider using '#align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_leₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
@@ -793,7 +793,7 @@ end PrincipalIdealDomain
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u2}} {M : Type.{u3}} {ι : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : Semiring.{u2} S] [_inst_3 : AddCommMonoid.{u3} M] [_inst_4 : Algebra.{u1, u2} R S _inst_1 _inst_2] [_inst_5 : Module.{u1, u3} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_6 : Module.{u2, u3} S M _inst_2 _inst_3] [_inst_7 : IsScalarTower.{u1, u2, u3} R S M (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2)))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_2))) (Algebra.toModule.{u1, u2} R S _inst_1 _inst_2 _inst_4))))) (SMulZeroClass.toHasSmul.{u2, u3} S M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_3))) (SMulWithZero.toSmulZeroClass.{u2, u3} S M (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S _inst_2)))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_3))) (MulActionWithZero.toSMulWithZero.{u2, u3} S M (Semiring.toMonoidWithZero.{u2} S _inst_2) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_3))) (Module.toMulActionWithZero.{u2, u3} S M _inst_2 _inst_3 _inst_6)))) (SMulZeroClass.toHasSmul.{u1, u3} R M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_3))) (SMulWithZero.toSmulZeroClass.{u1, u3} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_3))) (MulActionWithZero.toSMulWithZero.{u1, u3} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_3))) (Module.toMulActionWithZero.{u1, u3} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_5))))], (Function.Injective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) (algebraMap.{u1, u2} R S _inst_1 _inst_2 _inst_4))) -> (forall {v : ι -> M}, (LinearIndependent.{u4, u2, u3} ι S M v _inst_2 _inst_3 _inst_6) -> (LinearIndependent.{u4, u1, u3} ι R M v (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_5))
 but is expected to have type
-  forall {R : Type.{u4}} {S : Type.{u3}} {M : Type.{u2}} {ι : Type.{u1}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : Semiring.{u3} S] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Algebra.{u4, u3} R S _inst_1 _inst_2] [_inst_5 : Module.{u4, u2} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_6 : Module.{u3, u2} S M _inst_2 _inst_3] [_inst_7 : IsScalarTower.{u4, u3, u2} R S M (Algebra.toSMul.{u4, u3} R S _inst_1 _inst_2 _inst_4) (SMulZeroClass.toSMul.{u3, u2} S M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u3, u2} S M (MonoidWithZero.toZero.{u3} S (Semiring.toMonoidWithZero.{u3} S _inst_2)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u3, u2} S M (Semiring.toMonoidWithZero.{u3} S _inst_2) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u3, u2} S M _inst_2 _inst_3 _inst_6)))) (SMulZeroClass.toSMul.{u4, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u4, u2} R M (CommMonoidWithZero.toZero.{u4} R (CommSemiring.toCommMonoidWithZero.{u4} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u4, u2} R M (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u4, u2} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_5))))], (Function.Injective.{succ u4, succ u3} R S (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (RingHom.{u4, u3} R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u4 u3, u4, u3} (RingHom.{u4, u3} R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2)) R S (NonUnitalNonAssocSemiring.toMul.{u4} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u3, u4, u3} (RingHom.{u4, u3} R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u4 u3, u4, u3} (RingHom.{u4, u3} R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2)) R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2) (RingHom.instRingHomClassRingHom.{u4, u3} R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2))))) (algebraMap.{u4, u3} R S _inst_1 _inst_2 _inst_4))) -> (forall {v : ι -> M}, (LinearIndependent.{u1, u3, u2} ι S M v _inst_2 _inst_3 _inst_6) -> (LinearIndependent.{u1, u4, u2} ι R M v (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_5))
+  forall {R : Type.{u4}} {S : Type.{u3}} {M : Type.{u2}} {ι : Type.{u1}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : Semiring.{u3} S] [_inst_3 : AddCommMonoid.{u2} M] [_inst_4 : Algebra.{u4, u3} R S _inst_1 _inst_2] [_inst_5 : Module.{u4, u2} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_6 : Module.{u3, u2} S M _inst_2 _inst_3] [_inst_7 : IsScalarTower.{u4, u3, u2} R S M (Algebra.toSMul.{u4, u3} R S _inst_1 _inst_2 _inst_4) (SMulZeroClass.toSMul.{u3, u2} S M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u3, u2} S M (MonoidWithZero.toZero.{u3} S (Semiring.toMonoidWithZero.{u3} S _inst_2)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u3, u2} S M (Semiring.toMonoidWithZero.{u3} S _inst_2) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u3, u2} S M _inst_2 _inst_3 _inst_6)))) (SMulZeroClass.toSMul.{u4, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (SMulWithZero.toSMulZeroClass.{u4, u2} R M (CommMonoidWithZero.toZero.{u4} R (CommSemiring.toCommMonoidWithZero.{u4} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (MulActionWithZero.toSMulWithZero.{u4, u2} R M (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (Module.toMulActionWithZero.{u4, u2} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_5))))], (Function.Injective.{succ u4, succ u3} R S (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (RingHom.{u4, u3} R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u4 u3, u4, u3} (RingHom.{u4, u3} R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2)) R S (NonUnitalNonAssocSemiring.toMul.{u4} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u3, u4, u3} (RingHom.{u4, u3} R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u4 u3, u4, u3} (RingHom.{u4, u3} R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2)) R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2) (RingHom.instRingHomClassRingHom.{u4, u3} R S (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (Semiring.toNonAssocSemiring.{u3} S _inst_2))))) (algebraMap.{u4, u3} R S _inst_1 _inst_2 _inst_4))) -> (forall {v : ι -> M}, (LinearIndependent.{u1, u3, u2} ι S M v _inst_2 _inst_3 _inst_6) -> (LinearIndependent.{u1, u4, u2} ι R M v (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_5))
 Case conversion may be inaccurate. Consider using '#align linear_independent.restrict_scalars_algebras LinearIndependent.restrict_scalars_algebrasₓ'. -/
 /-- A set of linearly independent vectors in a module `M` over a semiring `S` is also linearly
 independent over a subring `R` of `K`. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/c89fe2d59ae06402c3f55f978016d1ada444f57e

@@ -71,7 +71,7 @@ open Submodule.IsPrincipal Submodule
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}}, (Basis.{u3, u1, u2} ι R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Preorder.toHasLt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N)], (Eq.{succ u1} R (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) _inst_1 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) N (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))
 but is expected to have type
-  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}}, (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) -> (forall {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} R (Ring.toAddCommGroup.{u2} R _inst_1)) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
+  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}}, (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) -> (forall {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.semilinearMapClass.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.semilinearMapClass.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.semilinearMapClass.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} R (Ring.toAddCommGroup.{u2} R _inst_1)) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.semilinearMapClass.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
 Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
     {ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
@@ -142,7 +142,7 @@ open Submodule.IsPrincipal
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N O) {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R 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(Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) => (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ψ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {N : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} {O : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} (hNO : LE.le.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) {ϕ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))}, (forall (ψ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ψ N))) -> (forall [_inst_6 : Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) 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(CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) ψ (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {N : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} {O : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} (hNO : LE.le.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) {ϕ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))}, (forall (ψ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ψ N))) -> (forall [_inst_6 : Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N)] (y : M) (yN : Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) y N), (Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => R) (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN))) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) (fun (_x : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} R R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) 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(CommRing.toRing.{u2} R _inst_1) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N) _inst_6)) -> (forall (ψ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))), Dvd.dvd.{u2} R (semigroupDvd.{u2} R (SemigroupWithZero.toSemigroup.{u2} R (NonUnitalSemiring.toSemigroupWithZero.{u2} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u2} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u2} R (CommRing.toNonUnitalCommRing.{u2} R _inst_1)))))) (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (CommRing.toRing.{u2} R _inst_1) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N) _inst_6) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M 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(AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) (fun (_x : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} R R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) ψ (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
 Case conversion may be inaccurate. Consider using '#align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvdₓ'. -/
 theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
     (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
@@ -585,7 +585,7 @@ theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M)
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)))) N O) -> (Sigma.{0, max u2 u3} Nat (fun (o : Nat) => Sigma.{0, max u2 u3} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 _inst_3 (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) 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_inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.semilinearMapClass.{u2, u2, u3, u3} R R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
 but is expected to have type
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))) N O) -> (Sigma.{0, max u3 u2} Nat (fun (o : Nat) => Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.comap.{u2, u2, u3, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))) N O) -> (Sigma.{0, max u3 u2} Nat (fun (o : Nat) => Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.comap.{u2, u2, u3, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.semilinearMapClass.{u2, u2, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
 Case conversion may be inaccurate. Consider using '#align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLEₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -69,7 +69,7 @@ open Submodule.IsPrincipal Submodule
 
 /- warning: eq_bot_of_generator_maximal_map_eq_zero -> eq_bot_of_generator_maximal_map_eq_zero is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}}, (Basis.{u3, u1, u2} ι R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Preorder.toLT.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N)], (Eq.{succ u1} R (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) _inst_1 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) N (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}}, (Basis.{u3, u1, u2} ι R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Preorder.toHasLt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N)], (Eq.{succ u1} R (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) _inst_1 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) N (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))
 but is expected to have type
   forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}}, (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) -> (forall {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} R (Ring.toAddCommGroup.{u2} R _inst_1)) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
 Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zeroₓ'. -/
@@ -89,7 +89,7 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
 
 /- warning: eq_bot_of_generator_maximal_submodule_image_eq_zero -> eq_bot_of_generator_maximal_submoduleImage_eq_zero is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}} {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3}, (Basis.{u3, u1, u2} ι R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) (Ring.toSemiring.{u1} R _inst_1) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O)) -> (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) N O) -> (forall {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Preorder.toLT.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N)], (Eq.{succ u1} R (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) _inst_1 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N) _inst_4) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) N (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}} {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3}, (Basis.{u3, u1, u2} ι R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) (Ring.toSemiring.{u1} R _inst_1) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O)) -> (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) N O) -> (forall {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Preorder.toHasLt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N)], (Eq.{succ u1} R (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) _inst_1 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N) _inst_4) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) N (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))
 but is expected to have type
   forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}} {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3}, (Basis.{u1, u2, u3} ι R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) (Ring.toSemiring.{u2} R _inst_1) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O)) -> (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.completeLattice.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3))))) N O) -> (forall {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
 Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_submodule_image_eq_zero eq_bot_of_generator_maximal_submoduleImage_eq_zeroₓ'. -/
@@ -140,7 +140,7 @@ open Submodule.IsPrincipal
 
 /- warning: generator_maximal_submodule_image_dvd -> generator_maximal_submoduleImage_dvd is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) 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R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Preorder.toLT.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (PartialOrder.toPreorder.{u1} 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O ψ N))) -> (forall [_inst_6 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ϕ N)] (y : M) (yN : Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, 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(coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ψ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) 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(Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ψ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
 but is expected to have type
   forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {N : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} {O : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} (hNO : LE.le.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) {ϕ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))}, (forall (ψ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ψ N))) -> (forall [_inst_6 : Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N)] (y : M) (yN : Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) y N), (Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => R) (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN))) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R 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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) ψ (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
 Case conversion may be inaccurate. Consider using '#align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvdₓ'. -/
@@ -182,7 +182,7 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
 
 /- warning: submodule.basis_of_pid_aux -> Submodule.basis_of_pid_aux is a dubious translation:
 lean 3 declaration is
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(CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8))) -> (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N M) -> (Exists.{succ u3} O (fun (y : O) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) (fun (H : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) => Exists.{succ u2} R (fun (a : R) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) (fun (hay : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (M' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (N' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') (fun (N'_le_M' : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) (fun (y_ortho_M' : forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} 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(SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z N') -> (Eq.{succ u3} O 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(SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y)) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O 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(Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) (fun (H : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) => Exists.{succ u2} R (fun (a : R) => Exists.{0} (Membership.Mem.{u3, u3} O 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(AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) (fun (hay : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (M' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (N' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') (fun (N'_le_M' : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) (fun (y_ortho_M' : forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z N') -> (Eq.{succ u3} O 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 but is expected to have type
   forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_6 : Finite.{succ u3} ι] {O : Type.{u2}} [_inst_7 : AddCommGroup.{u2} O] [_inst_8 : Module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7)] (M : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (N : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8), (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) 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O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1538 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} 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(instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_aux Submodule.basis_of_pid_auxₓ'. -/
@@ -391,7 +391,7 @@ theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
 
 /- warning: submodule.basis_of_pid_of_le -> Submodule.basisOfPidOfLE is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N O) -> (Basis.{u3, u1, u2} ι R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O)) -> (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) N) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N O) -> (Basis.{u3, u1, u2} ι R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O)) -> (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) N) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N)))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) N O) -> (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O)) -> (Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x N)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N)))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLEₓ'. -/
@@ -408,7 +408,7 @@ noncomputable def Submodule.basisOfPidOfLE {ι : Type _} [Finite ι] {N O : Subm
 
 /- warning: submodule.basis_of_pid_of_le_span -> Submodule.basisOfPidOfLESpan is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {b : ι -> M}, (LinearIndependent.{u3, u1, u2} ι R M b (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι b))) -> (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) N) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {b : ι -> M}, (LinearIndependent.{u3, u1, u2} ι R M b (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι b))) -> (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) N) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N))))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {b : ι -> M}, (LinearIndependent.{u3, u1, u2} ι R M b (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) -> (forall {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) N (Submodule.span.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι b))) -> (Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x N)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N))))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_of_le_span Submodule.basisOfPidOfLESpanₓ'. -/
@@ -546,7 +546,7 @@ structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type _) (n : ℕ) wher
 
 /- warning: submodule.exists_smith_normal_form_of_le -> Submodule.exists_smith_normal_form_of_le is a dubious translation:
 lean 3 declaration is
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 but is expected to have type
   forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] [_inst_6 : Finite.{succ u3} ι], (Basis.{u3, u2, u1} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (O : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5), (LE.le.{u1} (Submodule.{u2, u1} R M 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_inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) -> (Exists.{1} Nat (fun (n : Nat) => Exists.{1} Nat (fun (o : Nat) => Exists.{0} (LE.le.{0} Nat instLENat n o) (fun (hno : LE.le.{0} Nat instLENat n o) => Exists.{max (succ u2) (succ u1)} (Basis.{0, u2, u1} (Fin o) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M 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(x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin o) => LE.le.{0} (Fin o) (instLEFin o) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE n o hno) i)) (Submodule.smul.{u2, u2, u1} R R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O (Algebra.toSMul.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (IsScalarTower.left.{u2, u1} R M (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (MulActionWithZero.toMulAction.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) (a i) (FunLike.coe.{max (succ u2) (succ u1), 1, succ u1} (Basis.{0, u2, u1} (Fin o) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O)) (Fin o) (fun (_x : Fin o) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : Fin o) => Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) _x) (Basis.funLike.{0, u2, u1} (Fin o) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O)) bO (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin o) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (Fin o) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin o) (DistribLattice.toLattice.{0} (Fin o) (instDistribLattice.{0} (Fin o) (Fin.instLinearOrderFin o)))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (Fin o) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (DistribLattice.toLattice.{0} (Fin o) (instDistribLattice.{0} (Fin o) (Fin.instLinearOrderFin o))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (Fin o) (Fin.instLinearOrderFin n) (DistribLattice.toLattice.{0} (Fin o) (instDistribLattice.{0} (Fin o) (Fin.instLinearOrderFin o))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin o) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin o) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin o) => LE.le.{0} (Fin o) (instLEFin o) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE n o hno) i))))))))))))
 Case conversion may be inaccurate. Consider using '#align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_leₓ'. -/
@@ -583,7 +583,7 @@ theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M)
 
 /- warning: submodule.smith_normal_form_of_le -> Submodule.smithNormalFormOfLE is a dubious translation:
 lean 3 declaration is
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)))) N O) -> (Sigma.{0, max u2 u3} Nat (fun (o : Nat) => Sigma.{0, max u2 u3} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 _inst_3 (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.comap.{u2, u2, u3, u3, u3} R R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.semilinearMapClass.{u2, u2, u3, u3} R R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toHasLe.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)))) N O) -> (Sigma.{0, max u2 u3} Nat (fun (o : Nat) => Sigma.{0, max u2 u3} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 _inst_3 (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.comap.{u2, u2, u3, u3, u3} R R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.semilinearMapClass.{u2, u2, u3, u3} R R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
 but is expected to have type
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))) N O) -> (Sigma.{0, max u3 u2} Nat (fun (o : Nat) => Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.comap.{u2, u2, u3, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
 Case conversion may be inaccurate. Consider using '#align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLEₓ'. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -184,7 +184,7 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] [_inst_6 : Finite.{succ u1} ι] {O : Type.{u3}} [_inst_7 : AddCommGroup.{u3} O] [_inst_8 : Module.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)] (M : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (N : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8), (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) M) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8 M) (Submodule.module.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8 M)) -> (Ne.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) N (Bot.bot.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Submodule.hasBot.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8))) -> (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N M) -> (Exists.{succ u3} O (fun (y : O) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) (fun (H : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) => Exists.{succ u2} R (fun (a : R) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) (fun (hay : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (M' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (N' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') (fun (N'_le_M' : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) (fun (y_ortho_M' : forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} 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(SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z N') -> (Eq.{succ u3} O 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(MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y)) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O 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 but is expected to have type
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(Bot.bot.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.instBotSubmodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1567 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O 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O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_ay_ortho_N' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) 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(instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
+  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_6 : Finite.{succ u3} ι] {O : Type.{u2}} [_inst_7 : AddCommGroup.{u2} O] [_inst_8 : Module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7)] (M : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (N : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8), (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x M)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M) (Submodule.module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M)) -> (Ne.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) N (Bot.bot.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.instBotSubmodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1538 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} 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(instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_aux Submodule.basis_of_pid_auxₓ'. -/
 /-- The induction hypothesis of `submodule.basis_of_pid` and `submodule.smith_normal_form`.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/d4437c68c8d350fc9d4e95e1e174409db35e30d7

@@ -184,7 +184,7 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] [_inst_6 : Finite.{succ u1} ι] {O : Type.{u3}} [_inst_7 : AddCommGroup.{u3} O] [_inst_8 : Module.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)] (M : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (N : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8), (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) M) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8 M) (Submodule.module.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8 M)) -> (Ne.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) N (Bot.bot.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Submodule.hasBot.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8))) -> (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N M) -> (Exists.{succ u3} O (fun (y : O) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) (fun (H : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) => Exists.{succ u2} R (fun (a : R) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) (fun (hay : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (M' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (N' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') (fun (N'_le_M' : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) (fun (y_ortho_M' : forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z N') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y)) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) (fun (ay_ortho_N' : forall (c : R) (z : O), (Membership.Mem.{u3, u3} O 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(OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (fun (_x : RelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))) => (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (RelEmbedding.hasCoeToFun.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Fin.hasLe (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m' (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) hnm) i)))))))))))))))))))))
 but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_6 : Finite.{succ u3} ι] {O : Type.{u2}} [_inst_7 : AddCommGroup.{u2} O] [_inst_8 : Module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7)] (M : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (N : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8), (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x M)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M) (Submodule.module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M)) -> (Ne.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) N (Bot.bot.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.instBotSubmodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1575 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_ay_ortho_N' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R 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(Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m')) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
+  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_6 : Finite.{succ u3} ι] {O : Type.{u2}} [_inst_7 : AddCommGroup.{u2} O] [_inst_8 : Module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7)] (M : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (N : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8), (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x M)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M) (Submodule.module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M)) -> (Ne.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) N (Bot.bot.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.instBotSubmodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1567 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_ay_ortho_N' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} 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(Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m')) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 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(instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_aux Submodule.basis_of_pid_auxₓ'. -/
 /-- The induction hypothesis of `submodule.basis_of_pid` and `submodule.smith_normal_form`.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/08e1d8d4d989df3a6df86f385e9053ec8a372cc1

@@ -71,7 +71,7 @@ open Submodule.IsPrincipal Submodule
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}}, (Basis.{u3, u1, u2} ι R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Preorder.toLT.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N)], (Eq.{succ u1} R (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) _inst_1 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) N (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))
 but is expected to have type
-  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}}, (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) -> (forall {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} R (Ring.toAddCommGroup.{u2} R _inst_1)) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
+  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}}, (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) -> (forall {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} R (Ring.toAddCommGroup.{u2} R _inst_1)) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
 Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
     {ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
@@ -91,7 +91,7 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}} {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3}, (Basis.{u3, u1, u2} ι R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) (Ring.toSemiring.{u1} R _inst_1) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O)) -> (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) N O) -> (forall {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Preorder.toLT.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N)], (Eq.{succ u1} R (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) _inst_1 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N) _inst_4) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) N (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))
 but is expected to have type
-  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}} {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3}, (Basis.{u1, u2, u3} ι R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) (Ring.toSemiring.{u2} R _inst_1) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O)) -> (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.completeLattice.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3))))) N O) -> (forall {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
+  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}} {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3}, (Basis.{u1, u2, u3} ι R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) (Ring.toSemiring.{u2} R _inst_1) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O)) -> (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.completeLattice.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3))))) N O) -> (forall {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
 Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_submodule_image_eq_zero eq_bot_of_generator_maximal_submoduleImage_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
     (hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
@@ -142,7 +142,7 @@ open Submodule.IsPrincipal
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N O) {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Preorder.toLT.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ϕ N) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ψ N))) -> (forall [_inst_6 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ϕ N)] (y : M) (yN : Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, 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(NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (CommRing.toRing.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ϕ N) _inst_6) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) 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(Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) => (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ψ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {N : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} {O : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} (hNO : LE.le.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) {ϕ : LinearMap.{u2, u2, u1, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))}, (forall (ψ : LinearMap.{u2, u2, u1, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R 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u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) (fun (_x : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} R R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) ψ (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {N : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} {O : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} (hNO : LE.le.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) {ϕ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))}, (forall (ψ : LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ψ N))) -> (forall [_inst_6 : Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (LinearMap.submoduleImage.{u2, u1, u2} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) O ϕ N)] (y : M) (yN : Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) y N), (Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => R) (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN))) (FunLike.coe.{max (succ u2) (succ u1), succ u1, succ u2} (LinearMap.{u2, u2, u1, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R 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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) ψ (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
 Case conversion may be inaccurate. Consider using '#align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvdₓ'. -/
 theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
     (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
@@ -184,7 +184,7 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] [_inst_6 : Finite.{succ u1} ι] {O : Type.{u3}} [_inst_7 : AddCommGroup.{u3} O] [_inst_8 : Module.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)] (M : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (N : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8), (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) M) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8 M) (Submodule.module.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8 M)) -> (Ne.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) N (Bot.bot.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Submodule.hasBot.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8))) -> (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N M) -> (Exists.{succ u3} O (fun (y : O) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) (fun (H : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) => Exists.{succ u2} R (fun (a : R) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) (fun (hay : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (M' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (N' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') (fun (N'_le_M' : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) (fun (y_ortho_M' : forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} 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(SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z N') -> (Eq.{succ u3} O 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(MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y)) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O 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 but is expected to have type
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(Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1575 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O 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(Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x M)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M) (Submodule.module.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M)) bM (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (a : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) a) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) 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(HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
+  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_6 : Finite.{succ u3} ι] {O : Type.{u2}} [_inst_7 : AddCommGroup.{u2} O] [_inst_8 : Module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7)] (M : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (N : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8), (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x M)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M) (Submodule.module.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M)) -> (Ne.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) N (Bot.bot.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.instBotSubmodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1575 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_ay_ortho_N' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O 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Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m')) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_aux Submodule.basis_of_pid_auxₓ'. -/
 /-- The induction hypothesis of `submodule.basis_of_pid` and `submodule.smith_normal_form`.
 
@@ -334,7 +334,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι], (Basis.{u3, u1, u2} ι R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) -> (forall (N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5), Exists.{1} Nat (fun (n : Nat) => Nonempty.{max 1 (succ u1) (succ u2)} (Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) N) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N))))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι], (Basis.{u3, u2, u1} ι R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5), Exists.{1} Nat (fun (n : Nat) => Nonempty.{max (max (succ u1) (succ u2)) 1} (Basis.{0, u2, u1} (Fin n) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x N)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 N) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 N))))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι], (Basis.{u3, u2, u1} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5), Exists.{1} Nat (fun (n : Nat) => Nonempty.{max (max (succ u1) (succ u2)) 1} (Basis.{0, u2, u1} (Fin n) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x N)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 N) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 N))))
 Case conversion may be inaccurate. Consider using '#align submodule.nonempty_basis_of_pid Submodule.nonempty_basis_of_pidₓ'. -/
 /-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
@@ -378,7 +378,7 @@ noncomputable def Submodule.basisOfPid {ι : Type _} [Finite ι] (b : Basis ι R
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] (b : Basis.{u3, u1, u2} ι R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5), Eq.{succ (max u1 u2)} (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) (Submodule.basisOfPid.{u1, u2, u3} R _inst_1 _inst_2 _inst_3 M _inst_4 _inst_5 ι _inst_6 b (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (Sigma.mk.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R 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 but is expected to have type
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(CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Unique.instSubsingleton.{succ u1} (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (Submodule.uniqueBot.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) instIsEmptyFinOfNatNatInstOfNatNat))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5), Eq.{max (succ u2) (succ u1)} (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u2, u1} (Fin n) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) (Submodule.basisOfPid.{u2, u1, u3} R _inst_1 _inst_2 _inst_3 M _inst_4 _inst_5 ι _inst_6 b (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Sigma.mk.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u2, u1} (Fin n) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) (Basis.empty.{0, u2, u1} (Fin (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))) (Unique.instSubsingleton.{succ u1} (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)))) (Submodule.uniqueBot.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) instIsEmptyFinOfNatNatInstOfNatNat))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_bot Submodule.basisOfPid_botₓ'. -/
 theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
     Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ :=
@@ -393,7 +393,7 @@ theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N O) -> (Basis.{u3, u1, u2} ι R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O)) -> (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) N) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) N O) -> (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O)) -> (Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x N)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) N O) -> (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O)) -> (Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x N)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N)))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLEₓ'. -/
 /-- A submodule inside a free `R`-submodule of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
@@ -410,7 +410,7 @@ noncomputable def Submodule.basisOfPidOfLE {ι : Type _} [Finite ι] {N O : Subm
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {b : ι -> M}, (LinearIndependent.{u3, u1, u2} ι R M b (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι b))) -> (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) N) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {b : ι -> M}, (LinearIndependent.{u3, u1, u2} ι R M b (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) N (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι b))) -> (Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x N)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {b : ι -> M}, (LinearIndependent.{u3, u1, u2} ι R M b (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) -> (forall {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) N (Submodule.span.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι b))) -> (Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x N)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N))))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_of_le_span Submodule.basisOfPidOfLESpanₓ'. -/
 /-- A submodule inside the span of a linear independent family is a free `R`-module of finite rank,
 if `R` is a principal ideal domain. -/
@@ -426,7 +426,7 @@ variable {M}
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Fintype.{u1} ι] {s : ι -> M}, (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.span.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 (Set.range.{u3, succ u1} M ι s)) (Top.top.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.hasTop.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (SubNegMonoid.toAddMonoid.{u3} M (AddGroup.toSubNegMonoid.{u3} M (AddCommGroup.toAddGroup.{u3} M _inst_4))))) (SMulZeroClass.toHasSmul.{u2, u3} R M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (Module.toMulActionWithZero.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))], Sigma.{0, max u2 u3} Nat (fun (n : Nat) => Basis.{0, u2, u3} (Fin n) R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))
 but is expected to have type
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Fintype.{u1} ι] {s : ι -> M}, (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.span.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 (Set.range.{u3, succ u1} M ι s)) (Top.top.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.instTopSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (SMulZeroClass.toSMul.{u2, u3} R M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (Module.toMulActionWithZero.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))], Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.{0, u2, u3} (Fin n) R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Fintype.{u1} ι] {s : ι -> M}, (Eq.{succ u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.span.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 (Set.range.{u3, succ u1} M ι s)) (Top.top.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.instTopSubmodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (SMulZeroClass.toSMul.{u2, u3} R M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (Module.toMulActionWithZero.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))], Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.{0, u2, u3} (Fin n) R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))
 Case conversion may be inaccurate. Consider using '#align module.basis_of_finite_type_torsion_free Module.basisOfFiniteTypeTorsionFreeₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
 /-- A finite type torsion free module over a PID admits a basis. -/
@@ -489,7 +489,7 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι] {s : ι -> M}, (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.span.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 (Set.range.{u3, succ u1} M ι s)) (Top.top.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.hasTop.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (SubNegMonoid.toAddMonoid.{u3} M (AddGroup.toSubNegMonoid.{u3} M (AddCommGroup.toAddGroup.{u3} M _inst_4))))) (SMulZeroClass.toHasSmul.{u2, u3} R M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (Module.toMulActionWithZero.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))], Module.Free.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)
 but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Finite.{succ u3} ι] {s : ι -> M}, (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι s)) (Top.top.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.instTopSubmodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Module.Free.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
+  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Finite.{succ u3} ι] {s : ι -> M}, (Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.span.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι s)) (Top.top.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.instTopSubmodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Module.Free.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
 Case conversion may be inaccurate. Consider using '#align module.free_of_finite_type_torsion_free Module.free_of_finite_type_torsion_freeₓ'. -/
 theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs : span R (range s) = ⊤)
     [NoZeroSMulDivisors R M] : Module.Free R M :=
@@ -503,7 +503,7 @@ theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Module.Finite.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4))))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Module.Finite.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Module.Finite.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
 Case conversion may be inaccurate. Consider using '#align module.basis_of_finite_type_torsion_free' Module.basisOfFiniteTypeTorsionFree'ₓ'. -/
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree' [Module.Finite R M]
@@ -515,7 +515,7 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree' [Module.Finite R M]
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Module.Finite.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4))))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Module.Free.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] [_inst_6 : Module.Finite.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u2, u1} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))], Module.Free.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] [_inst_6 : Module.Finite.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u2, u1} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))], Module.Free.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5
 Case conversion may be inaccurate. Consider using '#align module.free_of_finite_type_torsion_free' Module.free_of_finite_type_torsion_free'ₓ'. -/
 theorem Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMulDivisors R M] :
     Module.Free R M :=
@@ -530,7 +530,7 @@ section SmithNormal
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)], (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) -> Type.{u3} -> Nat -> Sort.{max (succ u1) (succ u2) (succ u3)}
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] {_inst_3 : Type.{u2}} [M : AddCommGroup.{u2} _inst_3] [_inst_4 : Module.{u1, u2} R _inst_3 (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} _inst_3 M)], (Submodule.{u1, u2} R _inst_3 (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} _inst_3 M) _inst_4) -> Type.{u3} -> Nat -> Sort.{max (max (succ u1) (succ u2)) (succ u3)}
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] {_inst_3 : Type.{u2}} [M : AddCommGroup.{u2} _inst_3] [_inst_4 : Module.{u1, u2} R _inst_3 (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} _inst_3 M)], (Submodule.{u1, u2} R _inst_3 (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} _inst_3 M) _inst_4) -> Type.{u3} -> Nat -> Sort.{max (max (succ u1) (succ u2)) (succ u3)}
 Case conversion may be inaccurate. Consider using '#align basis.smith_normal_form Basis.SmithNormalFormₓ'. -/
 /-- A Smith normal form basis for a submodule `N` of a module `M` consists of
 bases for `M` and `N` such that the inclusion map `N → M` can be written as a
@@ -548,7 +548,7 @@ structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type _) (n : ℕ) wher
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)))) N O) -> (Exists.{1} Nat (fun (n : Nat) => Exists.{1} Nat (fun (o : Nat) => Exists.{0} (LE.le.{0} Nat Nat.hasLe n o) (fun (hno : LE.le.{0} Nat Nat.hasLe n o) => Exists.{max 1 (succ u2) (succ u3)} (Basis.{0, u2, u3} (Fin o) R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O)) (fun (bO : Basis.{0, u2, u3} (Fin o) R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O)) => Exists.{max 1 (succ u2) (succ u3)} (Basis.{0, u2, u3} (Fin n) R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) N) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 N) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 N)) (fun (bN : Basis.{0, u2, u3} (Fin n) R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M 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 but is expected to have type
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x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE n o hno) i)) (Submodule.smul.{u2, u2, u1} R R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O (Algebra.toSMul.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M 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n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin o) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (Fin o) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin o) (DistribLattice.toLattice.{0} (Fin o) (instDistribLattice.{0} (Fin o) (Fin.instLinearOrderFin o)))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (Fin o) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (DistribLattice.toLattice.{0} (Fin o) (instDistribLattice.{0} (Fin o) (Fin.instLinearOrderFin o))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (Fin o) (Fin.instLinearOrderFin n) (DistribLattice.toLattice.{0} (Fin o) (instDistribLattice.{0} (Fin o) (Fin.instLinearOrderFin o))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin o) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin o) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin o) => LE.le.{0} (Fin o) (instLEFin o) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE n o hno) i))))))))))))
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] [_inst_6 : Finite.{succ u3} ι], (Basis.{u3, u2, u1} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (O : Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5), (LE.le.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) -> (Exists.{1} Nat (fun (n : Nat) => Exists.{1} Nat (fun (o : Nat) => Exists.{0} (LE.le.{0} Nat instLENat n o) (fun (hno : LE.le.{0} Nat instLENat n o) => Exists.{max (succ u2) (succ u1)} (Basis.{0, u2, u1} (Fin o) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M 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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) _x) (Basis.funLike.{0, u2, u1} (Fin o) R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O)) bO (FunLike.coe.{1, 1, 1} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (fun (_x : Fin n) => (fun (x._@.Mathlib.Order.Hom.Lattice._hyg.494 : Fin n) => Fin o) _x) (InfHomClass.toFunLike.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (Fin o) (Lattice.toInf.{0} (Fin n) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n)))) (Lattice.toInf.{0} (Fin o) (DistribLattice.toLattice.{0} (Fin o) (instDistribLattice.{0} (Fin o) (Fin.instLinearOrderFin o)))) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (Fin o) (DistribLattice.toLattice.{0} (Fin n) (instDistribLattice.{0} (Fin n) (Fin.instLinearOrderFin n))) (DistribLattice.toLattice.{0} (Fin o) (instDistribLattice.{0} (Fin o) (Fin.instLinearOrderFin o))) (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin n) (Fin o) (instLEFin n) (instLEFin o)) (Fin n) (Fin o) (Fin.instLinearOrderFin n) (DistribLattice.toLattice.{0} (Fin o) (instDistribLattice.{0} (Fin o) (Fin.instLinearOrderFin o))) (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin n) (Fin o) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin n) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin n) => LE.le.{0} (Fin n) (instLEFin n) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin o) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin o) => LE.le.{0} (Fin o) (instLEFin o) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE n o hno) i))))))))))))
 Case conversion may be inaccurate. Consider using '#align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_leₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
@@ -585,7 +585,7 @@ theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M)
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)))) N O) -> (Sigma.{0, max u2 u3} Nat (fun (o : Nat) => Sigma.{0, max u2 u3} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 _inst_3 (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.comap.{u2, u2, u3, u3, u3} R R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.semilinearMapClass.{u2, u2, u3, u3} R R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
 but is expected to have type
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))) N O) -> (Sigma.{0, max u3 u2} Nat (fun (o : Nat) => Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.comap.{u2, u2, u3, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))) N O) -> (Sigma.{0, max u3 u2} Nat (fun (o : Nat) => Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.comap.{u2, u2, u3, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
 Case conversion may be inaccurate. Consider using '#align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLEₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
@@ -612,7 +612,7 @@ noncomputable def Submodule.smithNormalFormOfLE [Finite ι] (b : Basis ι R M) (
 lean 3 declaration is
   forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {M : Type.{u_3}} [_inst_4 : AddCommGroup.{u_3} M] [_inst_5 : Module.{u_2, u_3} R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4)] [_inst_6 : Finite.{succ u_1} ι], (Basis.{u_1, u_2, u_3} ι R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u_2, u_3} R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4) _inst_5), Sigma.{0, max u_2 u_3 u_1} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 _inst_3 M _inst_4 _inst_5 N ι n))
 but is expected to have type
-  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {M : Type.{u_3}} [_inst_4 : AddCommGroup.{u_3} M] [_inst_5 : Module.{u_2, u_3} R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4)] [_inst_6 : Finite.{succ u_1} ι], (Basis.{u_1, u_2, u_3} ι R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u_2, u_3} R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4) _inst_5), Sigma.{0, max (max u_1 u_3) u_2} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 M _inst_4 _inst_5 N ι n))
+  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {M : Type.{u_3}} [_inst_4 : AddCommGroup.{u_3} M] [_inst_5 : Module.{u_2, u_3} R M (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4)] [_inst_6 : Finite.{succ u_1} ι], (Basis.{u_1, u_2, u_3} ι R M (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u_2, u_3} R M (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4) _inst_5), Sigma.{0, max (max u_1 u_3) u_2} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 M _inst_4 _inst_5 N ι n))
 Case conversion may be inaccurate. Consider using '#align submodule.smith_normal_form Submodule.smithNormalFormₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
@@ -642,7 +642,7 @@ variable {S : Type _} [CommRing S] [IsDomain S] [Algebra R S]
 lean 3 declaration is
   forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {S : Type.{u_4}} [_inst_6 : CommRing.{u_4} S] [_inst_7 : IsDomain.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))] [_inst_8 : Algebra.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))] [_inst_9 : Fintype.{u_1} ι], (Basis.{u_1, u_2, u_4} ι R S (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_4} S (NonUnitalNonAssocRing.toAddCommGroup.{u_4} S (NonAssocRing.toNonUnitalNonAssocRing.{u_4} S (Ring.toNonAssocRing.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))), (Ne.{succ u_4} (Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) I (Bot.bot.{u_4} (Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) (Submodule.hasBot.{u_4, u_4} S S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Semiring.toModule.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))) -> (Basis.SmithNormalForm.{u_2, u_4, u_1} R _inst_1 _inst_2 _inst_3 S (NonUnitalNonAssocRing.toAddCommGroup.{u_4} S (NonAssocRing.toNonUnitalNonAssocRing.{u_4} S (Ring.toNonAssocRing.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8) (Submodule.restrictScalars.{u_2, u_4, u_4} R S S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8) (Semiring.toModule.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) (SMulZeroClass.toHasSmul.{u_2, u_4} R S (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u_2, u_4} R S (MulZeroClass.toHasZero.{u_2} R (MulZeroOneClass.toMulZeroClass.{u_2} R (MonoidWithZero.toMulZeroOneClass.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)))))) (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (MulActionWithZero.toSMulWithZero.{u_2, u_4} R S (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1))) (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (Module.toMulActionWithZero.{u_2, u_4} R S (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8))))) (Ideal.smithNormalForm._proof_1.{u_2, u_4} R _inst_1 S _inst_6 _inst_8) I) ι (Fintype.card.{u_1} ι _inst_9)))
 but is expected to have type
-  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {S : Type.{u_3}} [_inst_6 : CommRing.{u_3} S] [_inst_7 : IsDomain.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))] [_inst_8 : Algebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))] [_inst_9 : Fintype.{u_1} ι], (Basis.{u_1, u_2, u_3} ι R S (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonAssocRing.toNonUnitalNonAssocRing.{u_3} S (Ring.toNonAssocRing.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))), (Ne.{succ u_3} (Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) I (Bot.bot.{u_3} (Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) (Submodule.instBotSubmodule.{u_3, u_3} S S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Semiring.toModule.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)))))) -> (Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 S (Ring.toAddCommGroup.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (Submodule.restrictScalars.{u_2, u_3, u_3} R S S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (Semiring.toModule.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) (Algebra.toSMul.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (IsScalarTower.right.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) I) ι (Fintype.card.{u_1} ι _inst_9)))
+  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {S : Type.{u_3}} [_inst_6 : CommRing.{u_3} S] [_inst_7 : IsDomain.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))] [_inst_8 : Algebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))] [_inst_9 : Fintype.{u_1} ι], (Basis.{u_1, u_2, u_3} ι R S (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonAssocRing.toNonUnitalNonAssocRing.{u_3} S (Ring.toNonAssocRing.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))), (Ne.{succ u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))) I (Bot.bot.{u_3} (Ideal.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))) (Submodule.instBotSubmodule.{u_3, u_3} S S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))))) (Semiring.toModule.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)))))) -> (Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 S (Ring.toAddCommGroup.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) _inst_8) (Submodule.restrictScalars.{u_2, u_3, u_3} R S S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))))) (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) _inst_8) (Semiring.toModule.{u_3} S (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6))) (Algebra.toSMul.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) _inst_8) (IsScalarTower.right.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (CommSemiring.toSemiring.{u_3} S (CommRing.toCommSemiring.{u_3} S _inst_6)) _inst_8) I) ι (Fintype.card.{u_1} ι _inst_9)))
 Case conversion may be inaccurate. Consider using '#align ideal.smith_normal_form Ideal.smithNormalFormₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -669,7 +669,7 @@ variable [Finite ι]
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Exists.{max (succ u1) (succ u2) (succ u3)} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (fun (b' : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) => Exists.{max (succ u1) (succ u2)} (ι -> R) (fun (a : ι -> R) => Exists.{max (succ u1) (succ u2) (succ u3)} (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S 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(Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (SMulZeroClass.toHasSmul.{u2, u3} R S (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} R S (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S 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_inst_6)) _inst_8))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)))) ab' i)) (SMul.smul.{u2, u3} R S (SMulZeroClass.toHasSmul.{u2, u3} R S (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} R S (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R S (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (Module.toMulActionWithZero.{u2, u3} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))) (a i) (coeFn.{max (succ u1) (succ u2) (succ u3), max (succ u1) (succ u3)} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (fun (_x : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) => ι -> S) (FunLike.hasCoeToFun.{max (succ u1) (succ u2) (succ u3), succ u1, succ u3} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) ι (fun (_x : ι) => S) (Basis.funLike.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))) b' i)))))))
 but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι], (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))), (Ne.{succ u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))))) -> (Exists.{max (max (succ u3) (succ u2)) (succ u1)} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) (fun (b' : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) => Exists.{max (succ u3) (succ u2)} (ι -> R) (fun (a : ι -> R) => Exists.{max (max (succ u3) (succ u2)) (succ u1)} (Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) (fun (ab' : Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) => forall (i : ι), Eq.{succ u1} S (Subtype.val.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Set.{u1} S) (Set.instMembershipSet.{u1} S) x (SetLike.coe.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))) I)) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) _x) (Basis.funLike.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) ab' i)) (HSMul.hSMul.{u2, u1, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (instHSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (Algebra.toSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toRing.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) _inst_6)) _inst_8)) (a i) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) _x) (Basis.funLike.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) b' i)))))))
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι], (Basis.{u3, u2, u1} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))), (Ne.{succ u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)))))) -> (Exists.{max (max (succ u3) (succ u2)) (succ u1)} (Basis.{u3, u2, u1} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8)) (fun (b' : Basis.{u3, u2, u1} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8)) => Exists.{max (succ u3) (succ u2)} (ι -> R) (fun (a : ι -> R) => Exists.{max (max (succ u3) (succ u2)) (succ u1)} (Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => 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(CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8))) (fun (ab' : Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) x I)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8))) => forall (i : ι), Eq.{succ u1} S (Subtype.val.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Set.{u1} S) (Set.instMembershipSet.{u1} S) x (SetLike.coe.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)))) I)) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) x I)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8))) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) x I)) _x) (Basis.funLike.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) x I)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8))) ab' i)) (HSMul.hSMul.{u2, u1, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (instHSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (Algebra.toSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toCommSemiring.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) _inst_6)) _inst_8)) (a i) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8)) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) _x) (Basis.funLike.{u3, u2, u1} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8)) b' i)))))))
 Case conversion may be inaccurate. Consider using '#align ideal.exists_smith_normal_form Ideal.exists_smith_normal_formₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -701,7 +701,7 @@ theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)))
 but is expected to have type
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)))
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) _inst_8)))
 Case conversion may be inaccurate. Consider using '#align ideal.ring_basis Ideal.ringBasisₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -719,7 +719,7 @@ noncomputable def Ideal.ringBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I) (Submodule.module'.{u2, u3, u3} R S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (SMulZeroClass.toHasSmul.{u2, u3} R S (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} R S (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R S (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (Module.toMulActionWithZero.{u2, u3} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (Ideal.selfBasis._proof_1.{u2, u3} R _inst_1 S _inst_6 _inst_8))))
 but is expected to have type
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R (Subtype.{succ u3} S (fun (x : S) => Membership.mem.{u3, u3} S (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (SetLike.instMembership.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I) (Submodule.module'.{u2, u3, u3} R S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R (Subtype.{succ u3} S (fun (x : S) => Membership.mem.{u3, u3} S (Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))) (SetLike.instMembership.{u3, u3} (Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))))) x I)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u3, u3} S S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))) I) (Submodule.module'.{u2, u3, u3} R S S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))) I (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.toSMul.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) _inst_8))))
 Case conversion may be inaccurate. Consider using '#align ideal.self_basis Ideal.selfBasisₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -737,7 +737,7 @@ noncomputable def Ideal.selfBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> ι -> R)
 but is expected to have type
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> ι -> R)
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_6)))))) -> ι -> R)
 Case conversion may be inaccurate. Consider using '#align ideal.smith_coeffs Ideal.smithCoeffsₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -755,7 +755,7 @@ noncomputable def Ideal.smithCoeffs (b : Basis ι R S) (I : Ideal S) (hI : I ≠
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι] (b : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (hI : Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) (i : ι), Eq.{succ u3} S ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) S (HasLiftT.mk.{succ u3, succ u3} (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) S (CoeTCₓ.coe.{succ u3, succ u3} 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_inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} R S (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R S (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (Module.toMulActionWithZero.{u2, u3} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))) (Ideal.smithCoeffs.{u1, u2, u3} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (coeFn.{max (succ u1) (succ u2) (succ u3), max (succ u1) (succ u3)} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (fun (_x : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) => ι -> S) (FunLike.hasCoeToFun.{max (succ u1) (succ u2) (succ u3), succ u1, succ u3} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) ι (fun (_x : ι) => S) (Basis.funLike.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))) (Ideal.ringBasis.{u1, u2, u3} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI) i))
 but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) (I : Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (hI : Ne.{succ u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))))) (i : ι), Eq.{succ u1} S (Subtype.val.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Set.{u1} S) (Set.instMembershipSet.{u1} S) x (SetLike.coe.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))) I)) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) _x) 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_inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) (Ideal.selfBasis.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI) i)) (HSMul.hSMul.{u2, u1, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (instHSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (Algebra.toSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toRing.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) _inst_6)) _inst_8)) (Ideal.smithCoeffs.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) _x) (Basis.funLike.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) (Ideal.ringBasis.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI) i))
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8)) (I : Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (hI : Ne.{succ u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)))))) (i : ι), Eq.{succ u1} S (Subtype.val.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Set.{u1} S) (Set.instMembershipSet.{u1} S) x (SetLike.coe.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)))) I)) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) x I)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8))) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) x I)) _x) (Basis.funLike.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) x I)) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8))) (Ideal.selfBasis.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI) i)) (HSMul.hSMul.{u2, u1, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (instHSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (Algebra.toSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toCommSemiring.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) _inst_6)) _inst_8)) (Ideal.smithCoeffs.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8)) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) _x) (Basis.funLike.{u3, u2, u1} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8)) (Ideal.ringBasis.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI) i))
 Case conversion may be inaccurate. Consider using '#align ideal.self_basis_def Ideal.selfBasis_defₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -772,7 +772,7 @@ theorem Ideal.selfBasis_def (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι] (b : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (hI : Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) (i : ι), Ne.{succ u2} R (Ideal.smithCoeffs.{u1, u2, u3} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))
 but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) (I : Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (hI : Ne.{succ u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))))) (i : ι), Ne.{succ u2} R (Ideal.smithCoeffs.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2)))))
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) _inst_8)) (I : Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (hI : Ne.{succ u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_6)))))) (i : ι), Ne.{succ u2} R (Ideal.smithCoeffs.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2)))))
 Case conversion may be inaccurate. Consider using '#align ideal.smith_coeffs_ne_zero Ideal.smithCoeffs_ne_zeroₓ'. -/
 @[simp]
 theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) (i) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/9b2b58d6b14b895b2f375108e765cb47de71aebd

@@ -71,7 +71,7 @@ open Submodule.IsPrincipal Submodule
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}}, (Basis.{u3, u1, u2} ι R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Preorder.toLT.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N)], (Eq.{succ u1} R (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) _inst_1 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) N (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))
 but is expected to have type
-  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}}, (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) -> (forall {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)))))) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} R (Ring.toAddCommGroup.{u2} R _inst_1)) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
+  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}}, (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) -> (forall {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u2} R (Ring.toAddCommGroup.{u2} R _inst_1)) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
 Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
     {ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
@@ -91,7 +91,7 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}} {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3}, (Basis.{u3, u1, u2} ι R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) (Ring.toSemiring.{u1} R _inst_1) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O)) -> (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) N O) -> (forall {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Preorder.toLT.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N)], (Eq.{succ u1} R (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) _inst_1 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) O ϕ N) _inst_4) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) N (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))
 but is expected to have type
-  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}} {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3}, (Basis.{u1, u2, u3} ι R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) (Ring.toSemiring.{u2} R _inst_1) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O)) -> (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.completeLattice.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3))))) N O) -> (forall {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)))))) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) O ϕ N) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) O ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) O ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
+  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}} {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3}, (Basis.{u1, u2, u3} ι R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) (Ring.toSemiring.{u2} R _inst_1) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O)) -> (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.completeLattice.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3))))) N O) -> (forall {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1))))))) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) O ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
 Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_submodule_image_eq_zero eq_bot_of_generator_maximal_submoduleImage_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
     (hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
@@ -142,7 +142,7 @@ open Submodule.IsPrincipal
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N O) {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Preorder.toLT.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ϕ N) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ψ N))) -> (forall [_inst_6 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) O ϕ N)] (y : M) (yN : Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, 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(coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ψ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {N : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} {O : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} (hNO : LE.le.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))) N O) {ϕ : LinearMap.{u2, u2, u1, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} 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u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_1 (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))), Not (LT.lt.{u2} (Submodule.{u2, u2} R R 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(fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {N : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} {O : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} (hNO : LE.le.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) 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(fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} 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(AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u1, u2} R R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O)) R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5 O) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) ψ (Subtype.mk.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
 Case conversion may be inaccurate. Consider using '#align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvdₓ'. -/
 theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
     (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
@@ -184,7 +184,7 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] [_inst_6 : Finite.{succ u1} ι] {O : Type.{u3}} [_inst_7 : AddCommGroup.{u3} O] [_inst_8 : Module.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)] (M : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (N : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8), (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) M) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8 M) (Submodule.module.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8 M)) -> (Ne.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) N (Bot.bot.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Submodule.hasBot.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8))) -> (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N M) -> (Exists.{succ u3} O (fun (y : O) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) (fun (H : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) => Exists.{succ u2} R (fun (a : R) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) (fun (hay : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (M' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (N' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') (fun (N'_le_M' : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) (fun (y_ortho_M' : forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} 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(SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z N') -> (Eq.{succ u3} O 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 but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_6 : Finite.{succ u3} ι] {O : Type.{u2}} [_inst_7 : AddCommGroup.{u2} O] [_inst_8 : Module.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7)] (M : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (N : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8), (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x M)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M) (Submodule.module.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M)) -> (Ne.{succ u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) N (Bot.bot.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.instBotSubmodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1563 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_ay_ortho_N' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => forall (n' : Nat) (bN' : Basis.{0, u1, u2} (Fin n') R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x N')) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 N') (Submodule.module.{u1, u2} R O (Ring.toSemiring.{u1} 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Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
+  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_6 : Finite.{succ u3} ι] {O : Type.{u2}} [_inst_7 : AddCommGroup.{u2} O] [_inst_8 : Module.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7)] (M : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (N : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8), (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} O (fun (x : O) => Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) x M)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M) (Submodule.module.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8 M)) -> (Ne.{succ u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) N (Bot.bot.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.instBotSubmodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))) -> (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1575 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_ay_ortho_N' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, 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(instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))))) (Lattice.toInf.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m')) (LatticeHomClass.toInfHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (DistribLattice.toLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instDistribLattice.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (OrderHomClass.toLatticeHomClass.{0, 0, 0} (OrderEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLinearOrderFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin.instLatticeFinHAddNatInstHAddInstAddNatOfNat m') (RelEmbedding.instRelHomClassRelEmbedding.{0, 0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.680 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.680 x._@.Mathlib.Order.Hom.Basic._hyg.682) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.695 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) => LE.le.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (instLEFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) x._@.Mathlib.Order.Hom.Basic._hyg.695 x._@.Mathlib.Order.Hom.Basic._hyg.697))))) (Fin.castLE (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m' (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) hnm) i)))))))))))))))))))))
 Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_aux Submodule.basis_of_pid_auxₓ'. -/
 /-- The induction hypothesis of `submodule.basis_of_pid` and `submodule.smith_normal_form`.
 
@@ -642,7 +642,7 @@ variable {S : Type _} [CommRing S] [IsDomain S] [Algebra R S]
 lean 3 declaration is
   forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {S : Type.{u_4}} [_inst_6 : CommRing.{u_4} S] [_inst_7 : IsDomain.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))] [_inst_8 : Algebra.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))] [_inst_9 : Fintype.{u_1} ι], (Basis.{u_1, u_2, u_4} ι R S (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_4} S (NonUnitalNonAssocRing.toAddCommGroup.{u_4} S (NonAssocRing.toNonUnitalNonAssocRing.{u_4} S (Ring.toNonAssocRing.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))), (Ne.{succ u_4} (Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) I (Bot.bot.{u_4} (Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) (Submodule.hasBot.{u_4, u_4} S S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Semiring.toModule.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))) -> (Basis.SmithNormalForm.{u_2, u_4, u_1} R _inst_1 _inst_2 _inst_3 S (NonUnitalNonAssocRing.toAddCommGroup.{u_4} S (NonAssocRing.toNonUnitalNonAssocRing.{u_4} S (Ring.toNonAssocRing.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8) (Submodule.restrictScalars.{u_2, u_4, u_4} R S S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8) (Semiring.toModule.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) (SMulZeroClass.toHasSmul.{u_2, u_4} R S (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u_2, u_4} R S (MulZeroClass.toHasZero.{u_2} R (MulZeroOneClass.toMulZeroClass.{u_2} R (MonoidWithZero.toMulZeroOneClass.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)))))) (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (MulActionWithZero.toSMulWithZero.{u_2, u_4} R S (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1))) (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (Module.toMulActionWithZero.{u_2, u_4} R S (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8))))) (Ideal.smithNormalForm._proof_1.{u_2, u_4} R _inst_1 S _inst_6 _inst_8) I) ι (Fintype.card.{u_1} ι _inst_9)))
 but is expected to have type
-  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {S : Type.{u_3}} [_inst_6 : CommRing.{u_3} S] [_inst_7 : IsDomain.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))] [_inst_8 : Algebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))] [_inst_9 : Fintype.{u_1} ι], (Basis.{u_1, u_2, u_3} ι R S (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonAssocRing.toNonUnitalNonAssocRing.{u_3} S (Ring.toNonAssocRing.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) _inst_6 _inst_8)) -> (forall (I : Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))), (Ne.{succ u_3} (Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) I (Bot.bot.{u_3} (Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) (Submodule.instBotSubmodule.{u_3, u_3} S S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Semiring.toModule.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)))))) -> (Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 S (Ring.toAddCommGroup.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (Submodule.restrictScalars.{u_2, u_3, u_3} R S S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (Semiring.toModule.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) (Algebra.toSMul.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (IsScalarTower.right.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) I) ι (Fintype.card.{u_1} ι _inst_9)))
+  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {S : Type.{u_3}} [_inst_6 : CommRing.{u_3} S] [_inst_7 : IsDomain.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))] [_inst_8 : Algebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))] [_inst_9 : Fintype.{u_1} ι], (Basis.{u_1, u_2, u_3} ι R S (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonAssocRing.toNonUnitalNonAssocRing.{u_3} S (Ring.toNonAssocRing.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))), (Ne.{succ u_3} (Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) I (Bot.bot.{u_3} (Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) (Submodule.instBotSubmodule.{u_3, u_3} S S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Semiring.toModule.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)))))) -> (Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 S (Ring.toAddCommGroup.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (Submodule.restrictScalars.{u_2, u_3, u_3} R S S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (Semiring.toModule.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) (Algebra.toSMul.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (IsScalarTower.right.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) I) ι (Fintype.card.{u_1} ι _inst_9)))
 Case conversion may be inaccurate. Consider using '#align ideal.smith_normal_form Ideal.smithNormalFormₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -669,7 +669,7 @@ variable [Finite ι]
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Exists.{max (succ u1) (succ u2) (succ u3)} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (fun (b' : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) => Exists.{max (succ u1) (succ u2)} (ι -> R) (fun (a : ι -> R) => Exists.{max (succ u1) (succ u2) (succ u3)} (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I) (Submodule.module'.{u2, u3, u3} R S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (SMulZeroClass.toHasSmul.{u2, u3} R S (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} R S (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R S (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (Module.toMulActionWithZero.{u2, u3} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))) (fun (ab' : Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I) (Submodule.module'.{u2, u3, u3} R S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (SMulZeroClass.toHasSmul.{u2, u3} R S (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S 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_inst_6)) _inst_8))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)))) ab' i)) (SMul.smul.{u2, u3} R S (SMulZeroClass.toHasSmul.{u2, u3} R S (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} R S (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} S 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(Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))) (a i) (coeFn.{max (succ u1) (succ u2) (succ u3), max (succ u1) (succ u3)} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (fun (_x : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) => ι -> S) (FunLike.hasCoeToFun.{max (succ u1) (succ u2) (succ u3), succ u1, succ u3} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) ι (fun (_x : ι) => S) (Basis.funLike.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))) b' i)))))))
 but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι], (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) -> (forall (I : Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))), (Ne.{succ u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))))) -> (Exists.{max (max (succ u3) (succ u2)) (succ u1)} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) (fun (b' : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) => Exists.{max (succ u3) (succ u2)} (ι -> R) (fun (a : ι -> R) => Exists.{max (max (succ u3) (succ u2)) (succ u1)} (Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) 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_inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) => forall (i : ι), Eq.{succ u1} S (Subtype.val.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Set.{u1} S) (Set.instMembershipSet.{u1} S) x (SetLike.coe.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))) I)) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S 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(Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) ab' i)) (HSMul.hSMul.{u2, u1, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (instHSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (Algebra.toSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toRing.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) _inst_6)) _inst_8)) (a i) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) _x) (Basis.funLike.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) b' i)))))))
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι], (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))), (Ne.{succ u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))))) -> (Exists.{max (max (succ u3) (succ u2)) (succ u1)} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) (fun (b' : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) => Exists.{max (succ u3) (succ u2)} (ι -> R) (fun (a : ι -> R) => Exists.{max (max (succ u3) (succ u2)) (succ u1)} (Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S 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(CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) => forall (i : ι), Eq.{succ u1} S (Subtype.val.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Set.{u1} S) (Set.instMembershipSet.{u1} S) x (SetLike.coe.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))) I)) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) _x) (Basis.funLike.{u3, u2, u1} ι R (Subtype.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) ab' i)) (HSMul.hSMul.{u2, u1, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (instHSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (Algebra.toSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toRing.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) _inst_6)) _inst_8)) (a i) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) _x) (Basis.funLike.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) b' i)))))))
 Case conversion may be inaccurate. Consider using '#align ideal.exists_smith_normal_form Ideal.exists_smith_normal_formₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -701,7 +701,7 @@ theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)))
 but is expected to have type
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)))
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)))
 Case conversion may be inaccurate. Consider using '#align ideal.ring_basis Ideal.ringBasisₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -719,7 +719,7 @@ noncomputable def Ideal.ringBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I) (Submodule.module'.{u2, u3, u3} R S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (SMulZeroClass.toHasSmul.{u2, u3} R S (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} R S (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R S (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (Module.toMulActionWithZero.{u2, u3} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (Ideal.selfBasis._proof_1.{u2, u3} R _inst_1 S _inst_6 _inst_8))))
 but is expected to have type
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R (Subtype.{succ u3} S (fun (x : S) => Membership.mem.{u3, u3} S (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (SetLike.instMembership.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I) (Submodule.module'.{u2, u3, u3} R S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R (Subtype.{succ u3} S (fun (x : S) => Membership.mem.{u3, u3} S (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (SetLike.instMembership.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I) (Submodule.module'.{u2, u3, u3} R S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))
 Case conversion may be inaccurate. Consider using '#align ideal.self_basis Ideal.selfBasisₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -737,7 +737,7 @@ noncomputable def Ideal.selfBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> ι -> R)
 but is expected to have type
-  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> ι -> R)
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> ι -> R)
 Case conversion may be inaccurate. Consider using '#align ideal.smith_coeffs Ideal.smithCoeffsₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -755,7 +755,7 @@ noncomputable def Ideal.smithCoeffs (b : Basis ι R S) (I : Ideal S) (hI : I ≠
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι] (b : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (hI : Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) (i : ι), Eq.{succ u3} S ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) S (HasLiftT.mk.{succ u3, succ u3} (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) S (CoeTCₓ.coe.{succ u3, succ u3} (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) S (coeBase.{succ u3, succ u3} (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) S (coeSubtype.{succ u3} S (fun (x : S) => Membership.Mem.{u3, u3} S (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (SetLike.hasMem.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) x I))))) (coeFn.{max (succ u1) (succ u2) (succ u3), max (succ u1) (succ u3)} (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} 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(CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (fun (_x : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) => ι -> S) (FunLike.hasCoeToFun.{max (succ u1) (succ u2) (succ u3), succ u1, succ u3} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) ι (fun (_x : ι) => S) (Basis.funLike.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))) (Ideal.ringBasis.{u1, u2, u3} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI) i))
 but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) (I : Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (hI : Ne.{succ u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))))) (i : ι), Eq.{succ u1} S (Subtype.val.{succ u1} S (fun (x : S) => Membership.mem.{u1, u1} S (Set.{u1} S) 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(CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) 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_inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) _x) (Basis.funLike.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) (Ideal.ringBasis.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI) i))
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) (I : Ideal.{u1} S 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_inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I) (Submodule.module'.{u2, u1, u1} R S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8))) (Ideal.selfBasis.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI) i)) (HSMul.hSMul.{u2, u1, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (instHSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (Algebra.toSMul.{u2, u1} R ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) (CommRing.toRing.{u1} ((fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) _inst_6)) _inst_8)) (Ideal.smithCoeffs.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) _x) (Basis.funLike.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) (Ideal.ringBasis.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI) i))
 Case conversion may be inaccurate. Consider using '#align ideal.self_basis_def Ideal.selfBasis_defₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -772,7 +772,7 @@ theorem Ideal.selfBasis_def (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
 lean 3 declaration is
   forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι] (b : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (hI : Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) (i : ι), Ne.{succ u2} R (Ideal.smithCoeffs.{u1, u2, u3} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))
 but is expected to have type
-  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) (I : Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (hI : Ne.{succ u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))))) (i : ι), Ne.{succ u2} R (Ideal.smithCoeffs.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2)))))
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) _inst_8)) (I : Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (hI : Ne.{succ u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))))) (i : ι), Ne.{succ u2} R (Ideal.smithCoeffs.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2)))))
 Case conversion may be inaccurate. Consider using '#align ideal.smith_coeffs_ne_zero Ideal.smithCoeffs_ne_zeroₓ'. -/
 @[simp]
 theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) (i) :

mathlib commit https://github.com/leanprover-community/mathlib/commit/52932b3a083d4142e78a15dc928084a22fea9ba0

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 
 ! This file was ported from Lean 3 source module linear_algebra.free_module.pid
-! leanprover-community/mathlib commit 210657c4ea4a4a7b234392f70a3a2a83346dfa90
+! leanprover-community/mathlib commit 86d1873c01a723aba6788f0b9051ae3d23b4c1c3
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.RingTheory.Finiteness
 
 /-! # Free modules over PID
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 A free `R`-module `M` is a module with a basis over `R`,
 equivalently it is an `R`-module linearly equivalent to `ι →₀ R` for some `ι`.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/730c6d4cab72b9d84fcfb9e95e8796e9cd8f40ba

@@ -64,6 +64,12 @@ variable {ι : Type _} (b : Basis ι R M)
 
 open Submodule.IsPrincipal Submodule
 
+/- warning: eq_bot_of_generator_maximal_map_eq_zero -> eq_bot_of_generator_maximal_map_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}}, (Basis.{u3, u1, u2} ι R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3} {ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))}, (forall (ψ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))), Not (LT.lt.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Preorder.toLT.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)))))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N)], (Eq.{succ u1} R (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) _inst_1 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) ϕ N) _inst_4) (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))))))))) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) N (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] {ι : Type.{u1}}, (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) -> (forall {N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3} {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)}, (forall (ψ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)))))) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (Submodule.map.{u2, u2, u3, u2, max u2 u3} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (RingHomSurjective.ids.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M R (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u2} R R M R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) _inst_3 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R 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+Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
     {ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
     (hgen : generator (N.map ϕ) = (0 : R)) : N = ⊥ :=
@@ -78,6 +84,12 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
       ⟨x, hx, rfl⟩
 #align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zero
 
+/- warning: eq_bot_of_generator_maximal_submodule_image_eq_zero -> eq_bot_of_generator_maximal_submoduleImage_eq_zero is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3))))) N O) -> (forall {ϕ : LinearMap.{u2, u2, u3, u2} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)) x O)) R (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R 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(Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 O) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)), Not (LT.lt.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (Preorder.toLT.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (PartialOrder.toPreorder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1)))))) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) O ϕ N) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) O ψ N))) -> (forall [_inst_4 : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) O ϕ N)], (Eq.{succ u2} R (Submodule.IsPrincipal.generator.{u2, u2} R R (Ring.toAddCommGroup.{u2} R _inst_1) _inst_1 (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) (LinearMap.submoduleImage.{u2, u3, u2} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u2} R _inst_1) O ϕ N) _inst_4) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R _inst_1)))))) -> (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) _inst_3)))))
+Case conversion may be inaccurate. Consider using '#align eq_bot_of_generator_maximal_submodule_image_eq_zero eq_bot_of_generator_maximal_submoduleImage_eq_zeroₓ'. -/
 theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
     (hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (hgen : generator (ϕ.submoduleImage N) = 0) : N = ⊥ :=
@@ -101,6 +113,7 @@ variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal Set Submodule
 
+#print dvd_generator_iff /-
 theorem dvd_generator_iff {I : Ideal R} [I.IsPrincipal] {x : R} (hx : x ∈ I) :
     x ∣ generator I ↔ I = Ideal.span {x} :=
   by
@@ -108,6 +121,7 @@ theorem dvd_generator_iff {I : Ideal R} [I.IsPrincipal] {x : R} (hx : x ∈ I) :
   erw [Ideal.span_singleton_eq_span_singleton, ← dvd_dvd_iff_associated, ← mem_iff_generator_dvd]
   exact ⟨fun h => ⟨hx, h⟩, fun h => h.2⟩
 #align dvd_generator_iff dvd_generator_iff
+-/
 
 end IsDomain
 
@@ -121,6 +135,12 @@ variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal
 
+/- warning: generator_maximal_submodule_image_dvd -> generator_maximal_submoduleImage_dvd is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) 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(coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) ψ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O) y (hNO y yN)))))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] {N : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} {O : Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5} (hNO : LE.le.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5) 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+Case conversion may be inaccurate. Consider using '#align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvdₓ'. -/
 theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
     (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (y : M) (yN : y ∈ N)
@@ -157,6 +177,12 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
   · exact subset_span (mem_insert _ _)
 #align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvd
 
+/- warning: submodule.basis_of_pid_aux -> Submodule.basis_of_pid_aux is a dubious translation:
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(Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) (fun (H : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) y M) => Exists.{succ u2} R (fun (a : R) => Exists.{0} (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) (fun (hay : Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y) N) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (M' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) M' M) => Exists.{succ u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (fun (N' : Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) (fun (H : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' N) => Exists.{0} (LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') (fun (N'_le_M' : LE.le.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (Preorder.toLE.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.partialOrder.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) c y) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) (fun (y_ortho_M' : forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z M') -> (Eq.{succ u3} O (HAdd.hAdd.{u3, u3, u3} O O O (instHAdd.{u3} O (AddZeroClass.toHasAdd.{u3} O (AddMonoid.toAddZeroClass.{u3} O (SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7)))))) (SMul.smul.{u2, u3} R O (SMulZeroClass.toHasSmul.{u2, u3} R O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} 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(SubNegMonoid.toAddMonoid.{u3} O (AddGroup.toSubNegMonoid.{u3} O (AddCommGroup.toAddGroup.{u3} O _inst_7))))))))) -> (Eq.{succ u2} R c (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.Mem.{u3, u3} O (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8) O (Submodule.setLike.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)) z N') -> (Eq.{succ u3} O 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(SMulWithZero.toSmulZeroClass.{u2, u3} R O (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R O (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O (AddCommMonoid.toAddMonoid.{u3} O (AddCommGroup.toAddCommMonoid.{u3} O _inst_7)))) (Module.toMulActionWithZero.{u2, u3} R O (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} O _inst_7) _inst_8)))) a y)) z) (OfNat.ofNat.{u3} O 0 (OfNat.mk.{u3} O 0 (Zero.zero.{u3} O (AddZeroClass.toHasZero.{u3} O (AddMonoid.toAddZeroClass.{u3} O 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R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N M) -> (Exists.{succ u2} O (fun (y : O) => And (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) y M) (Exists.{succ u1} R (fun (a : R) => Exists.{0} (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) (fun (x._@.Mathlib.LinearAlgebra.FreeModule.PID._hyg.1563 : Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y) N) => Exists.{succ u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (M' : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) M' M) (Exists.{succ u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (fun (N' : Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) => And (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' N) (Exists.{0} (LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') (fun (_N'_le_M' : LE.le.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Preorder.toLE.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (Submodule.completeLattice.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) N' M') => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) (fun (_y_ortho_M' : forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z M') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c y) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))))) => Exists.{0} (forall (c : R) (z : O), (Membership.mem.{u2, u2} O (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8) O (Submodule.setLike.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8)) z N') -> (Eq.{succ u2} O (HAdd.hAdd.{u2, u2, u2} O O O (instHAdd.{u2} O (AddZeroClass.toAdd.{u2} O (AddMonoid.toAddZeroClass.{u2} O (SubNegMonoid.toAddMonoid.{u2} O (AddGroup.toSubNegMonoid.{u2} O (AddCommGroup.toAddGroup.{u2} O _inst_7)))))) (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) c (HSMul.hSMul.{u1, u2, u2} R O O (instHSMul.{u1, u2} R O (SMulZeroClass.toSMul.{u1, u2} R O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R O (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R O (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7))))) (Module.toMulActionWithZero.{u1, u2} R O (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} O _inst_7) _inst_8))))) a y)) z) (OfNat.ofNat.{u2} O 0 (Zero.toOfNat0.{u2} O (NegZeroClass.toZero.{u2} O (SubNegZeroMonoid.toNegZeroClass.{u2} O (SubtractionMonoid.toSubNegZeroMonoid.{u2} O (SubtractionCommMonoid.toSubtractionMonoid.{u2} O (AddCommGroup.toDivisionAddCommMonoid.{u2} O _inst_7)))))))) -> (Eq.{succ u1} R c (OfNat.ofNat.{u1} R 0 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+Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_aux Submodule.basis_of_pid_auxₓ'. -/
 /-- The induction hypothesis of `submodule.basis_of_pid` and `submodule.smith_normal_form`.
 
 Basically, it says: let `N ≤ M` be a pair of submodules, then we can find a pair of
@@ -301,6 +327,12 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
     simp only [Fin.cons_succ, coe_of_le, h i]
 #align submodule.basis_of_pid_aux Submodule.basis_of_pid_aux
 
+/- warning: submodule.nonempty_basis_of_pid -> Submodule.nonempty_basis_of_pid is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align submodule.nonempty_basis_of_pid Submodule.nonempty_basis_of_pidₓ'. -/
 /-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
 
@@ -327,6 +359,7 @@ theorem Submodule.nonempty_basis_of_pid {ι : Type _} [Finite ι] (b : Basis ι
   exact ⟨n' + 1, ⟨bN⟩⟩
 #align submodule.nonempty_basis_of_pid Submodule.nonempty_basis_of_pid
 
+#print Submodule.basisOfPid /-
 /-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
 
@@ -336,7 +369,14 @@ noncomputable def Submodule.basisOfPid {ι : Type _} [Finite ι] (b : Basis ι R
     (N : Submodule R M) : Σn : ℕ, Basis (Fin n) R N :=
   ⟨_, (N.nonempty_basis_of_pid b).choose_spec.some⟩
 #align submodule.basis_of_pid Submodule.basisOfPid
+-/
 
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_bot Submodule.basisOfPid_botₓ'. -/
 theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
     Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ :=
   by
@@ -346,27 +386,45 @@ theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
   exact Sigma.eq rfl (Basis.eq_of_apply_eq <| finZeroElim)
 #align submodule.basis_of_pid_bot Submodule.basisOfPid_bot
 
+/- warning: submodule.basis_of_pid_of_le -> Submodule.basisOfPidOfLE is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)))) N O) -> (Basis.{u3, u1, u2} ι R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) O) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O)) -> (Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) N) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) N O) -> (Basis.{u3, u1, u2} ι R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x O)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 O)) -> (Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x N)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N)))
+Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLEₓ'. -/
 /-- A submodule inside a free `R`-submodule of finite rank is also a free `R`-module of finite rank,
 if `R` is a principal ideal domain.
 
 See also the stronger version `submodule.smith_normal_form_of_le`.
 -/
-noncomputable def Submodule.basisOfPidOfLe {ι : Type _} [Finite ι] {N O : Submodule R M}
+noncomputable def Submodule.basisOfPidOfLE {ι : Type _} [Finite ι] {N O : Submodule R M}
     (hNO : N ≤ O) (b : Basis ι R O) : Σn : ℕ, Basis (Fin n) R N :=
   let ⟨n, bN'⟩ := Submodule.basisOfPid b (N.comap O.Subtype)
   ⟨n, bN'.map (Submodule.comapSubtypeEquivOfLe hNO)⟩
-#align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLe
-
+#align submodule.basis_of_pid_of_le Submodule.basisOfPidOfLE
+
+/- warning: submodule.basis_of_pid_of_le_span -> Submodule.basisOfPidOfLESpan is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] {ι : Type.{u3}} [_inst_6 : Finite.{succ u3} ι] {b : ι -> M}, (LinearIndependent.{u3, u1, u2} ι R M b (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) -> (forall {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5}, (LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))) N (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι b))) -> (Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)) x N)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 N))))
+Case conversion may be inaccurate. Consider using '#align submodule.basis_of_pid_of_le_span Submodule.basisOfPidOfLESpanₓ'. -/
 /-- A submodule inside the span of a linear independent family is a free `R`-module of finite rank,
 if `R` is a principal ideal domain. -/
-noncomputable def Submodule.basisOfPidOfLeSpan {ι : Type _} [Finite ι] {b : ι → M}
+noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι → M}
     (hb : LinearIndependent R b) {N : Submodule R M} (le : N ≤ Submodule.span R (Set.range b)) :
     Σn : ℕ, Basis (Fin n) R N :=
-  Submodule.basisOfPidOfLe le (Basis.span hb)
-#align submodule.basis_of_pid_of_le_span Submodule.basisOfPidOfLeSpan
+  Submodule.basisOfPidOfLE le (Basis.span hb)
+#align submodule.basis_of_pid_of_le_span Submodule.basisOfPidOfLESpan
 
 variable {M}
 
+/- warning: module.basis_of_finite_type_torsion_free -> Module.basisOfFiniteTypeTorsionFree is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Fintype.{u1} ι] {s : ι -> M}, (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.span.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 (Set.range.{u3, succ u1} M ι s)) (Top.top.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.hasTop.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (SubNegMonoid.toAddMonoid.{u3} M (AddGroup.toSubNegMonoid.{u3} M (AddCommGroup.toAddGroup.{u3} M _inst_4))))) (SMulZeroClass.toHasSmul.{u2, u3} R M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (Module.toMulActionWithZero.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))], Sigma.{0, max u2 u3} Nat (fun (n : Nat) => Basis.{0, u2, u3} (Fin n) R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))
+but is expected to have type
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Fintype.{u1} ι] {s : ι -> M}, (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.span.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 (Set.range.{u3, succ u1} M ι s)) (Top.top.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.instTopSubmodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (SMulZeroClass.toSMul.{u2, u3} R M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_4))))) (Module.toMulActionWithZero.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))], Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.{0, u2, u3} (Fin n) R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))
+Case conversion may be inaccurate. Consider using '#align module.basis_of_finite_type_torsion_free Module.basisOfFiniteTypeTorsionFreeₓ'. -/
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
@@ -419,11 +477,17 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M
         _ ∈ N := N.smul_mem _ (ha' i)
         
     -- Since a submodule of a free `R`-module is free, we get that `A • M` is free
-    obtain ⟨n, b : Basis (Fin n) R φ.range⟩ := Submodule.basisOfPidOfLe this sI_basis
+    obtain ⟨n, b : Basis (Fin n) R φ.range⟩ := Submodule.basisOfPidOfLE this sI_basis
     -- hence `M` is free.
     exact ⟨n, b.map ψ.symm⟩
 #align module.basis_of_finite_type_torsion_free Module.basisOfFiniteTypeTorsionFree
 
+/- warning: module.free_of_finite_type_torsion_free -> Module.free_of_finite_type_torsion_free is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι] {s : ι -> M}, (Eq.{succ u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.span.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 (Set.range.{u3, succ u1} M ι s)) (Top.top.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.hasTop.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (SubNegMonoid.toAddMonoid.{u3} M (AddGroup.toSubNegMonoid.{u3} M (AddCommGroup.toAddGroup.{u3} M _inst_4))))) (SMulZeroClass.toHasSmul.{u2, u3} R M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u2, u3} R M (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)))) (Module.toMulActionWithZero.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))], Module.Free.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)
+but is expected to have type
+  forall {ι : Type.{u3}} {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Finite.{succ u3} ι] {s : ι -> M}, (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5 (Set.range.{u2, succ u3} M ι s)) (Top.top.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) (Submodule.instTopSubmodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))) -> (forall [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Module.Free.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
+Case conversion may be inaccurate. Consider using '#align module.free_of_finite_type_torsion_free Module.free_of_finite_type_torsion_freeₓ'. -/
 theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs : span R (range s) = ⊤)
     [NoZeroSMulDivisors R M] : Module.Free R M :=
   by
@@ -432,12 +496,24 @@ theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs :
   exact Module.Free.of_basis b
 #align module.free_of_finite_type_torsion_free Module.free_of_finite_type_torsion_free
 
+/- warning: module.basis_of_finite_type_torsion_free' -> Module.basisOfFiniteTypeTorsionFree' is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Module.Finite.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4))))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Sigma.{0, max u1 u2} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Module.Finite.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_4))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Sigma.{0, max u2 u1} Nat (fun (n : Nat) => Basis.{0, u1, u2} (Fin n) R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5)
+Case conversion may be inaccurate. Consider using '#align module.basis_of_finite_type_torsion_free' Module.basisOfFiniteTypeTorsionFree'ₓ'. -/
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree' [Module.Finite R M]
     [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M :=
   Module.basisOfFiniteTypeTorsionFree Module.Finite.exists_fin.choose_spec.choose_spec
 #align module.basis_of_finite_type_torsion_free' Module.basisOfFiniteTypeTorsionFree'
 
+/- warning: module.free_of_finite_type_torsion_free' -> Module.free_of_finite_type_torsion_free' is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)] [_inst_6 : Module.Finite.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_4))))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5))))], Module.Free.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u1}} [_inst_4 : AddCommGroup.{u1} M] [_inst_5 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4)] [_inst_6 : Module.Finite.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5] [_inst_7 : NoZeroSMulDivisors.{u2, u1} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulZeroClass.toSMul.{u2, u1} R M (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (NegZeroClass.toZero.{u1} M (SubNegZeroMonoid.toNegZeroClass.{u1} M (SubtractionMonoid.toSubNegZeroMonoid.{u1} M (SubtractionCommMonoid.toSubtractionMonoid.{u1} M (AddCommGroup.toDivisionAddCommMonoid.{u1} M _inst_4))))) (Module.toMulActionWithZero.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5))))], Module.Free.{u2, u1} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_4) _inst_5
+Case conversion may be inaccurate. Consider using '#align module.free_of_finite_type_torsion_free' Module.free_of_finite_type_torsion_free'ₓ'. -/
 theorem Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMulDivisors R M] :
     Module.Free R M :=
   by
@@ -447,6 +523,12 @@ theorem Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMul
 
 section SmithNormal
 
+/- warning: basis.smith_normal_form -> Basis.SmithNormalForm is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {M : Type.{u2}} [_inst_4 : AddCommGroup.{u2} M] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4)], (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_4) _inst_5) -> Type.{u3} -> Nat -> Sort.{max (succ u1) (succ u2) (succ u3)}
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] {_inst_3 : Type.{u2}} [M : AddCommGroup.{u2} _inst_3] [_inst_4 : Module.{u1, u2} R _inst_3 (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} _inst_3 M)], (Submodule.{u1, u2} R _inst_3 (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} _inst_3 M) _inst_4) -> Type.{u3} -> Nat -> Sort.{max (max (succ u1) (succ u2)) (succ u3)}
+Case conversion may be inaccurate. Consider using '#align basis.smith_normal_form Basis.SmithNormalFormₓ'. -/
 /-- A Smith normal form basis for a submodule `N` of a module `M` consists of
 bases for `M` and `N` such that the inclusion map `N → M` can be written as a
 (rectangular) matrix with `a` along the diagonal: in Smith normal form. -/
@@ -459,6 +541,12 @@ structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type _) (n : ℕ) wher
   snf : ∀ i, (bN i : M) = a i • bM (f i)
 #align basis.smith_normal_form Basis.SmithNormalForm
 
+/- warning: submodule.exists_smith_normal_form_of_le -> Submodule.exists_smith_normal_form_of_le is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) 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+Case conversion may be inaccurate. Consider using '#align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_leₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
@@ -490,6 +578,12 @@ theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M)
   exact ⟨_, _, hmn, bM, bN, as, has⟩
 #align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_le
 
+/- warning: submodule.smith_normal_form_of_le -> Submodule.smithNormalFormOfLE is a dubious translation:
+lean 3 declaration is
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(Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.semilinearMapClass.{u2, u2, u3, u3} R R (coeSort.{succ u3, succ (succ u3)} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) O) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
+but is expected to have type
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {M : Type.{u3}} [_inst_4 : AddCommGroup.{u3} M] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4)] [_inst_6 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (O : Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5), (LE.le.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Preorder.toLE.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (Submodule.completeLattice.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5))))) N O) -> (Sigma.{0, max u3 u2} Nat (fun (o : Nat) => Sigma.{0, max u3 u2} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u2, u3, 0} R _inst_1 _inst_2 (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (Submodule.comap.{u2, u2, u3, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) (Submodule.addCommGroup.{u2, u3} R M (CommRing.toRing.{u2} R _inst_1) _inst_4 _inst_5 O)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (LinearMap.{u2, u2, u3, u3} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u3} R R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5) M (Submodule.setLike.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5)) x O)) M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) (Submodule.module.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) _inst_5 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Submodule.subtype.{u2, u3} R M (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_4) _inst_5 O) N) (Fin o) n))))
+Case conversion may be inaccurate. Consider using '#align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLEₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
@@ -499,7 +593,7 @@ need to map `N` into a submodule of `O`.
 
 This is a strengthening of `submodule.basis_of_pid_of_le`.
 -/
-noncomputable def Submodule.smithNormalFormOfLe [Finite ι] (b : Basis ι R M) (N O : Submodule R M)
+noncomputable def Submodule.smithNormalFormOfLE [Finite ι] (b : Basis ι R M) (N O : Submodule R M)
     (N_le_O : N ≤ O) : Σo n : ℕ, Basis.SmithNormalForm (N.comap O.Subtype) (Fin o) n :=
   by
   choose n o hno bO bN a snf using N.exists_smith_normal_form_of_le b O N_le_O
@@ -509,8 +603,14 @@ noncomputable def Submodule.smithNormalFormOfLe [Finite ι] (b : Basis ι R M) (
   ext
   simp only [snf, Basis.map_apply, Submodule.comapSubtypeEquivOfLe_symm_apply,
     Submodule.coe_smul_of_tower, RelEmbedding.coe_toEmbedding]
-#align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLe
-
+#align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLE
+
+/- warning: submodule.smith_normal_form -> Submodule.smithNormalForm is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {M : Type.{u_3}} [_inst_4 : AddCommGroup.{u_3} M] [_inst_5 : Module.{u_2, u_3} R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4)] [_inst_6 : Finite.{succ u_1} ι], (Basis.{u_1, u_2, u_3} ι R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u_2, u_3} R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4) _inst_5), Sigma.{0, max u_2 u_3 u_1} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 _inst_3 M _inst_4 _inst_5 N ι n))
+but is expected to have type
+  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {M : Type.{u_3}} [_inst_4 : AddCommGroup.{u_3} M] [_inst_5 : Module.{u_2, u_3} R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4)] [_inst_6 : Finite.{succ u_1} ι], (Basis.{u_1, u_2, u_3} ι R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4) _inst_5) -> (forall (N : Submodule.{u_2, u_3} R M (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_3} M _inst_4) _inst_5), Sigma.{0, max (max u_1 u_3) u_2} Nat (fun (n : Nat) => Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 M _inst_4 _inst_5 N ι n))
+Case conversion may be inaccurate. Consider using '#align submodule.smith_normal_form Submodule.smithNormalFormₓ'. -/
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.
@@ -522,7 +622,7 @@ an ideal is the same as the dimension of the whole ring.
 -/
 noncomputable def Submodule.smithNormalForm [Finite ι] (b : Basis ι R M) (N : Submodule R M) :
     Σn : ℕ, Basis.SmithNormalForm N ι n :=
-  let ⟨m, n, bM, bN, f, a, snf⟩ := N.smithNormalFormOfLe b ⊤ le_top
+  let ⟨m, n, bM, bN, f, a, snf⟩ := N.smithNormalFormOfLE b ⊤ le_top
   let bM' := bM.map (LinearEquiv.ofTop _ rfl)
   let e := bM'.indexEquiv b
   ⟨n, bM'.reindex e, bN.map (comapSubtypeEquivOfLe le_top), f.trans e.toEmbedding, a, fun i => by
@@ -535,6 +635,12 @@ section Ideal
 
 variable {S : Type _} [CommRing S] [IsDomain S] [Algebra R S]
 
+/- warning: ideal.smith_normal_form -> Ideal.smithNormalForm is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {S : Type.{u_4}} [_inst_6 : CommRing.{u_4} S] [_inst_7 : IsDomain.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))] [_inst_8 : Algebra.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))] [_inst_9 : Fintype.{u_1} ι], (Basis.{u_1, u_2, u_4} ι R S (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u_4} S (NonUnitalNonAssocRing.toAddCommGroup.{u_4} S (NonAssocRing.toNonUnitalNonAssocRing.{u_4} S (Ring.toNonAssocRing.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))), (Ne.{succ u_4} (Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) I (Bot.bot.{u_4} (Ideal.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) (Submodule.hasBot.{u_4, u_4} S S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Semiring.toModule.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))) -> (Basis.SmithNormalForm.{u_2, u_4, u_1} R _inst_1 _inst_2 _inst_3 S (NonUnitalNonAssocRing.toAddCommGroup.{u_4} S (NonAssocRing.toNonUnitalNonAssocRing.{u_4} S (Ring.toNonAssocRing.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8) (Submodule.restrictScalars.{u_2, u_4, u_4} R S S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8) (Semiring.toModule.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))) (SMulZeroClass.toHasSmul.{u_2, u_4} R S (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u_2, u_4} R S (MulZeroClass.toHasZero.{u_2} R (MulZeroOneClass.toMulZeroClass.{u_2} R (MonoidWithZero.toMulZeroOneClass.{u_2} R (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)))))) (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (MulActionWithZero.toSMulWithZero.{u_2, u_4} R S (Semiring.toMonoidWithZero.{u_2} R (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1))) (AddZeroClass.toHasZero.{u_4} S (AddMonoid.toAddZeroClass.{u_4} S (AddCommMonoid.toAddMonoid.{u_4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)))))))) (Module.toMulActionWithZero.{u_2, u_4} R S (CommSemiring.toSemiring.{u_2} R (CommRing.toCommSemiring.{u_2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_4} S (Semiring.toNonAssocSemiring.{u_4} S (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6))))) (Algebra.toModule.{u_2, u_4} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_4} S (CommRing.toRing.{u_4} S _inst_6)) _inst_8))))) (Ideal.smithNormalForm._proof_1.{u_2, u_4} R _inst_1 S _inst_6 _inst_8) I) ι (Fintype.card.{u_1} ι _inst_9)))
+but is expected to have type
+  forall {ι : Type.{u_1}} {R : Type.{u_2}} [_inst_1 : CommRing.{u_2} R] [_inst_2 : IsDomain.{u_2} R (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u_2} R (CommRing.toRing.{u_2} R _inst_1)] {S : Type.{u_3}} [_inst_6 : CommRing.{u_3} S] [_inst_7 : IsDomain.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))] [_inst_8 : Algebra.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))] [_inst_9 : Fintype.{u_1} ι], (Basis.{u_1, u_2, u_3} ι R S (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u_3} S (NonAssocRing.toNonUnitalNonAssocRing.{u_3} S (Ring.toNonAssocRing.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) _inst_6 _inst_8)) -> (forall (I : Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))), (Ne.{succ u_3} (Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) I (Bot.bot.{u_3} (Ideal.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) (Submodule.instBotSubmodule.{u_3, u_3} S S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Semiring.toModule.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)))))) -> (Basis.SmithNormalForm.{u_2, u_3, u_1} R _inst_1 _inst_2 S (Ring.toAddCommGroup.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (Submodule.restrictScalars.{u_2, u_3, u_3} R S S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u_3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u_3} S (Semiring.toNonAssocSemiring.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))))) (Ring.toSemiring.{u_2} R (CommRing.toRing.{u_2} R _inst_1)) (Algebra.toModule.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (Semiring.toModule.{u_3} S (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6))) (Algebra.toSMul.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) (IsScalarTower.right.{u_2, u_3} R S (CommRing.toCommSemiring.{u_2} R _inst_1) (Ring.toSemiring.{u_3} S (CommRing.toRing.{u_3} S _inst_6)) _inst_8) I) ι (Fintype.card.{u_1} ι _inst_9)))
+Case conversion may be inaccurate. Consider using '#align ideal.smith_normal_form Ideal.smithNormalFormₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -556,6 +662,12 @@ noncomputable def Ideal.smithNormalForm [Fintype ι] (b : Basis ι R S) (I : Ide
 
 variable [Finite ι]
 
+/- warning: ideal.exists_smith_normal_form -> Ideal.exists_smith_normal_form is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Exists.{max (succ u1) (succ u2) (succ u3)} (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) 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(CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))) b' i)))))))
+but is expected to have type
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι], (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) 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(x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) i) _inst_6)) _inst_8)) (a i) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) _x) (Basis.funLike.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) b' i)))))))
+Case conversion may be inaccurate. Consider using '#align ideal.exists_smith_normal_form Ideal.exists_smith_normal_formₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -582,6 +694,12 @@ theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I
           Submodule.restrictScalarsEquiv_apply, Basis.coe_reindex]⟩
 #align ideal.exists_smith_normal_form Ideal.exists_smith_normal_form
 
+/- warning: ideal.ring_basis -> Ideal.ringBasis is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)))
+but is expected to have type
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)))
+Case conversion may be inaccurate. Consider using '#align ideal.ring_basis Ideal.ringBasisₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -594,6 +712,12 @@ noncomputable def Ideal.ringBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   (Ideal.exists_smith_normal_form b I hI).some
 #align ideal.ring_basis Ideal.ringBasis
 
+/- warning: ideal.self_basis -> Ideal.selfBasis is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R (coeSort.{succ u3, succ (succ u3)} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) I) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I) (Submodule.module'.{u2, u3, u3} R S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (SMulZeroClass.toHasSmul.{u2, u3} R S (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} R S (MulZeroClass.toHasZero.{u2} R (MulZeroOneClass.toMulZeroClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R S (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddZeroClass.toHasZero.{u3} S (AddMonoid.toAddZeroClass.{u3} S (AddCommMonoid.toAddMonoid.{u3} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))))) (Module.toMulActionWithZero.{u2, u3} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (Ideal.selfBasis._proof_1.{u2, u3} R _inst_1 S _inst_6 _inst_8))))
+but is expected to have type
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> (Basis.{u1, u2, u3} ι R (Subtype.{succ u3} S (fun (x : S) => Membership.mem.{u3, u3} S (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (SetLike.instMembership.{u3, u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) S (Submodule.setLike.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) x I)) (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Submodule.addCommMonoid.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I) (Submodule.module'.{u2, u3, u3} R S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Algebra.toSMul.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8) (IsScalarTower.right.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8))))
+Case conversion may be inaccurate. Consider using '#align ideal.self_basis Ideal.selfBasisₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -606,6 +730,12 @@ noncomputable def Ideal.selfBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose_spec.some
 #align ideal.self_basis Ideal.selfBasis
 
+/- warning: ideal.smith_coeffs -> Ideal.smithCoeffs is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> ι -> R)
+but is expected to have type
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι], (Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) -> (forall (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))), (Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.instBotSubmodule.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) -> ι -> R)
+Case conversion may be inaccurate. Consider using '#align ideal.smith_coeffs Ideal.smithCoeffsₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -618,6 +748,12 @@ noncomputable def Ideal.smithCoeffs (b : Basis ι R S) (I : Ideal S) (hI : I ≠
   (Ideal.exists_smith_normal_form b I hI).choose_spec.some
 #align ideal.smith_coeffs Ideal.smithCoeffs
 
+/- warning: ideal.self_basis_def -> Ideal.selfBasis_def is a dubious translation:
+lean 3 declaration is
+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι] (b : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (I : Ideal.{u3} S (Ring.toSemiring.{u3} S 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+but is expected to have type
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) 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_inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), succ u3, succ u1} (Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) ι (fun (_x : ι) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : ι) => S) _x) (Basis.funLike.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) (Ideal.ringBasis.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI) i))
+Case conversion may be inaccurate. Consider using '#align ideal.self_basis_def Ideal.selfBasis_defₓ'. -/
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -629,6 +765,12 @@ theorem Ideal.selfBasis_def (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose_spec.choose_spec
 #align ideal.self_basis_def Ideal.selfBasis_def
 
+/- warning: ideal.smith_coeffs_ne_zero -> Ideal.smithCoeffs_ne_zero is a dubious translation:
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+  forall {ι : Type.{u1}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u3}} [_inst_6 : CommRing.{u3} S] [_inst_7 : IsDomain.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_8 : Algebra.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))] [_inst_9 : Finite.{succ u1} ι] (b : Basis.{u1, u2, u3} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} S (NonUnitalNonAssocRing.toAddCommGroup.{u3} S (NonAssocRing.toNonUnitalNonAssocRing.{u3} S (Ring.toNonAssocRing.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Algebra.toModule.{u2, u3} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) _inst_8)) (I : Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (hI : Ne.{succ u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) I (Bot.bot.{u3} (Ideal.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))) (Submodule.hasBot.{u3, u3} S S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6))))) (Semiring.toModule.{u3} S (Ring.toSemiring.{u3} S (CommRing.toRing.{u3} S _inst_6)))))) (i : ι), Ne.{succ u2} R (Ideal.smithCoeffs.{u1, u2, u3} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (OfNat.ofNat.{u2} R 0 (OfNat.mk.{u2} R 0 (Zero.zero.{u2} R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))
+but is expected to have type
+  forall {ι : Type.{u3}} {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] [_inst_2 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_1)] {S : Type.{u1}} [_inst_6 : CommRing.{u1} S] [_inst_7 : IsDomain.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_8 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))] [_inst_9 : Finite.{succ u3} ι] (b : Basis.{u3, u2, u1} ι R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_8)) (I : Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (hI : Ne.{succ u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) I (Bot.bot.{u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))) (Submodule.instBotSubmodule.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_6)))))) (i : ι), Ne.{succ u2} R (Ideal.smithCoeffs.{u3, u2, u1} ι R _inst_1 _inst_2 _inst_3 S _inst_6 _inst_7 _inst_8 _inst_9 b I hI i) (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2)))))
+Case conversion may be inaccurate. Consider using '#align ideal.smith_coeffs_ne_zero Ideal.smithCoeffs_ne_zeroₓ'. -/
 @[simp]
 theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) (i) :
     Ideal.smithCoeffs b I hI i ≠ 0 := by
@@ -644,6 +786,12 @@ end SmithNormal
 
 end PrincipalIdealDomain
 
+/- warning: linear_independent.restrict_scalars_algebras -> LinearIndependent.restrict_scalars_algebras is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align linear_independent.restrict_scalars_algebras LinearIndependent.restrict_scalars_algebrasₓ'. -/
 /-- A set of linearly independent vectors in a module `M` over a semiring `S` is also linearly
 independent over a subring `R` of `K`. -/
 theorem LinearIndependent.restrict_scalars_algebras {R S M ι : Type _} [CommSemiring R] [Semiring S]

mathlib commit https://github.com/leanprover-community/mathlib/commit/730c6d4cab72b9d84fcfb9e95e8796e9cd8f40ba

@@ -508,7 +508,7 @@ noncomputable def Submodule.smithNormalFormOfLe [Finite ι] (b : Basis ι R M) (
       fun i => _⟩
   ext
   simp only [snf, Basis.map_apply, Submodule.comapSubtypeEquivOfLe_symm_apply,
-    Submodule.coe_smul_of_tower, RelEmbedding.coeFn_toEmbedding]
+    Submodule.coe_smul_of_tower, RelEmbedding.coe_toEmbedding]
 #align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLe
 
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free

mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65

@@ -182,9 +182,9 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
                   ∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
                     ∃ (hnm : n' + 1 ≤ m' + 1)(bM : Basis (Fin (m' + 1)) R M),
                       ∀ (as : Fin n' → R)
-                        (h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLe hn'm' i) : O)),
+                        (h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)),
                         ∃ as' : Fin (n' + 1) → R,
-                          ∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLe hnm i) : O) :=
+                          ∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O) :=
   by
   -- Let `ϕ` be a maximal projection of `M` onto `R`, in the sense that there is
   -- no `ψ` whose image of `N` is larger than `ϕ`'s image of `N`.
@@ -295,9 +295,9 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   intro i
   rw [Basis.coe_mkFinConsOfLe, Basis.coe_mkFinConsOfLe]
   refine' Fin.cases _ (fun i => _) i
-  · simp only [Fin.cons_zero, Fin.castLe_zero]
+  · simp only [Fin.cons_zero, Fin.castLE_zero]
     exact a_smul_y'.symm
-  · rw [Fin.castLe_succ]
+  · rw [Fin.castLE_succ]
     simp only [Fin.cons_succ, coe_of_le, h i]
 #align submodule.basis_of_pid_aux Submodule.basis_of_pid_aux
 
@@ -471,7 +471,7 @@ This is a strengthening of `submodule.basis_of_pid_of_le`.
 theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M) (N O : Submodule R M)
     (N_le_O : N ≤ O) :
     ∃ (n o : ℕ)(hno : n ≤ o)(bO : Basis (Fin o) R O)(bN : Basis (Fin n) R N)(a : Fin n → R),
-      ∀ i, (bN i : M) = a i • bO (Fin.castLe hno i) :=
+      ∀ i, (bN i : M) = a i • bO (Fin.castLE hno i) :=
   by
   cases nonempty_fintype ι
   revert N
@@ -504,7 +504,7 @@ noncomputable def Submodule.smithNormalFormOfLe [Finite ι] (b : Basis ι R M) (
   by
   choose n o hno bO bN a snf using N.exists_smith_normal_form_of_le b O N_le_O
   refine'
-    ⟨o, n, bO, bN.map (comap_subtype_equiv_of_le N_le_O).symm, (Fin.castLe hno).toEmbedding, a,
+    ⟨o, n, bO, bN.map (comap_subtype_equiv_of_le N_le_O).symm, (Fin.castLE hno).toEmbedding, a,
       fun i => _⟩
   ext
   simp only [snf, Basis.map_apply, Submodule.comapSubtypeEquivOfLe_symm_apply,

mathlib commit https://github.com/leanprover-community/mathlib/commit/88fcb83fe7996142dfcfe7368d31304a9adc874a

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 
 ! This file was ported from Lean 3 source module linear_algebra.free_module.pid
-! leanprover-community/mathlib commit f62c15c01a5409b31b97a82d79a12980be4eff35
+! leanprover-community/mathlib commit 210657c4ea4a4a7b234392f70a3a2a83346dfa90
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -65,22 +65,21 @@ variable {ι : Type _} (b : Basis ι R M)
 open Submodule.IsPrincipal Submodule
 
 theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
-    {ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, N.map ϕ ≤ N.map ψ → N.map ψ = N.map ϕ)
-    [(N.map ϕ).IsPrincipal] (hgen : generator (N.map ϕ) = (0 : R)) : N = ⊥ :=
+    {ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
+    (hgen : generator (N.map ϕ) = (0 : R)) : N = ⊥ :=
   by
   rw [Submodule.eq_bot_iff]
   intro x hx
   refine' b.ext_elem fun i => _
   rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
   rw [LinearEquiv.map_zero, Finsupp.zero_apply]
-  exact (Submodule.eq_bot_iff _).mp (hϕ (Finsupp.lapply i ∘ₗ ↑b.repr) bot_le) _ ⟨x, hx, rfl⟩
+  exact
+    (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _
+      ⟨x, hx, rfl⟩
 #align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zero
 
 theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
-    (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
-    (hϕ :
-      ∀ ψ : O →ₗ[R] R,
-        ϕ.submoduleImage N ≤ ψ.submoduleImage N → ψ.submoduleImage N = ϕ.submoduleImage N)
+    (hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (hgen : generator (ϕ.submoduleImage N) = 0) : N = ⊥ :=
   by
   rw [Submodule.eq_bot_iff]
@@ -88,7 +87,7 @@ theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M}
   refine' congr_arg coe (show (⟨x, hNO hx⟩ : O) = 0 from b.ext_elem fun i => _)
   rw [(eq_bot_iff_generator_eq_zero _).mpr hgen] at hϕ
   rw [LinearEquiv.map_zero, Finsupp.zero_apply]
-  refine' (Submodule.eq_bot_iff _).mp (hϕ (Finsupp.lapply i ∘ₗ ↑b.repr) bot_le) _ _
+  refine' (Submodule.eq_bot_iff _).mp (not_bot_lt_iff.1 <| hϕ (Finsupp.lapply i ∘ₗ ↑b.repr)) _ _
   exact (LinearMap.mem_submoduleImage_of_le hNO).mpr ⟨x, hx, rfl⟩
 #align eq_bot_of_generator_maximal_submodule_image_eq_zero eq_bot_of_generator_maximal_submoduleImage_eq_zero
 
@@ -123,9 +122,7 @@ variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
 open Submodule.IsPrincipal
 
 theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
-    (hϕ :
-      ∀ ψ : O →ₗ[R] R,
-        ϕ.submoduleImage N ≤ ψ.submoduleImage N → ψ.submoduleImage N = ϕ.submoduleImage N)
+    (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (y : M) (yN : y ∈ N)
     (ϕy_eq : ϕ ⟨y, hNO yN⟩ = generator (ϕ.submoduleImage N)) (ψ : O →ₗ[R] R) :
     generator (ϕ.submoduleImage N) ∣ ψ ⟨y, hNO yN⟩ :=
@@ -154,7 +151,7 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
   refine'
     le_antisymm (this.trans (le_of_eq _)) (ideal.span_singleton_le_span_singleton.mpr d_dvd_left)
   rw [span_singleton_generator]
-  refine' hϕ ψ' (le_trans _ this)
+  apply (le_trans _ this).eq_of_not_gt (hϕ ψ')
   rw [← span_singleton_generator (ϕ.submodule_image N)]
   exact ideal.span_singleton_le_span_singleton.mpr d_dvd_left
   · exact subset_span (mem_insert _ _)
@@ -191,10 +188,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   by
   -- Let `ϕ` be a maximal projection of `M` onto `R`, in the sense that there is
   -- no `ψ` whose image of `N` is larger than `ϕ`'s image of `N`.
-  have :
-    ∃ ϕ : M →ₗ[R] R,
-      ∀ ψ : M →ₗ[R] R,
-        ϕ.submoduleImage N ≤ ψ.submoduleImage N → ψ.submoduleImage N = ϕ.submoduleImage N :=
+  have : ∃ ϕ : M →ₗ[R] R, ∀ ψ : M →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N :=
     by
     obtain ⟨P, P_eq, P_max⟩ :=
       set_has_maximal_iff_noetherian.mpr (inferInstance : IsNoetherian R R) _

mathlib commit https://github.com/leanprover-community/mathlib/commit/62e8311c791f02c47451bf14aa2501048e7c2f33

@@ -4,11 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 
 ! This file was ported from Lean 3 source module linear_algebra.free_module.pid
-! leanprover-community/mathlib commit b8e1f0f706714622ed3081a0ce1ea6cdc5e17371
+! leanprover-community/mathlib commit f62c15c01a5409b31b97a82d79a12980be4eff35
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.LinearAlgebra.Dimension
+import Mathbin.LinearAlgebra.FreeModule.Basic
 import Mathbin.RingTheory.PrincipalIdealDomain
 import Mathbin.RingTheory.Finiteness
 
@@ -373,8 +374,8 @@ noncomputable def Submodule.basisOfPidOfLeSpan {ι : Type _} [Finite ι] {b : ι
 variable {M}
 
 /- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
-/-- A finite type torsion free module over a PID is free. -/
-noncomputable def Module.freeOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
+/-- A finite type torsion free module over a PID admits a basis. -/
+noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
     (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M := by
   classical
     -- We define `N` as the submodule spanned by a maximal linear independent subfamily of `s`
@@ -427,13 +428,28 @@ noncomputable def Module.freeOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
     obtain ⟨n, b : Basis (Fin n) R φ.range⟩ := Submodule.basisOfPidOfLe this sI_basis
     -- hence `M` is free.
     exact ⟨n, b.map ψ.symm⟩
-#align module.free_of_finite_type_torsion_free Module.freeOfFiniteTypeTorsionFree
+#align module.basis_of_finite_type_torsion_free Module.basisOfFiniteTypeTorsionFree
 
-/-- A finite type torsion free module over a PID is free. -/
-noncomputable def Module.freeOfFiniteTypeTorsionFree' [Module.Finite R M] [NoZeroSMulDivisors R M] :
-    Σn : ℕ, Basis (Fin n) R M :=
-  Module.freeOfFiniteTypeTorsionFree Module.Finite.exists_fin.choose_spec.choose_spec
-#align module.free_of_finite_type_torsion_free' Module.freeOfFiniteTypeTorsionFree'
+theorem Module.free_of_finite_type_torsion_free [Finite ι] {s : ι → M} (hs : span R (range s) = ⊤)
+    [NoZeroSMulDivisors R M] : Module.Free R M :=
+  by
+  cases nonempty_fintype ι
+  obtain ⟨n, b⟩ : Σn, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree hs
+  exact Module.Free.of_basis b
+#align module.free_of_finite_type_torsion_free Module.free_of_finite_type_torsion_free
+
+/-- A finite type torsion free module over a PID admits a basis. -/
+noncomputable def Module.basisOfFiniteTypeTorsionFree' [Module.Finite R M]
+    [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M :=
+  Module.basisOfFiniteTypeTorsionFree Module.Finite.exists_fin.choose_spec.choose_spec
+#align module.basis_of_finite_type_torsion_free' Module.basisOfFiniteTypeTorsionFree'
+
+theorem Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMulDivisors R M] :
+    Module.Free R M :=
+  by
+  obtain ⟨n, b⟩ : Σn, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree'
+  exact Module.Free.of_basis b
+#align module.free_of_finite_type_torsion_free' Module.free_of_finite_type_torsion_free'
 
 section SmithNormal
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/4c586d291f189eecb9d00581aeb3dd998ac34442

@@ -372,7 +372,7 @@ noncomputable def Submodule.basisOfPidOfLeSpan {ι : Type _} [Finite ι] {b : ι
 
 variable {M}
 
-/- ./././Mathport/Syntax/Translate/Basic.lean:628:2: warning: expanding binder collection (i «expr ∉ » I) -/
+/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (i «expr ∉ » I) -/
 /-- A finite type torsion free module over a PID is free. -/
 noncomputable def Module.freeOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
     (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M := by
@@ -497,7 +497,7 @@ noncomputable def Submodule.smithNormalFormOfLe [Finite ι] (b : Basis ι R M) (
     ⟨o, n, bO, bN.map (comap_subtype_equiv_of_le N_le_O).symm, (Fin.castLe hno).toEmbedding, a,
       fun i => _⟩
   ext
-  simp only [snf, Basis.map_apply, Submodule.comapSubtypeEquivOfLe_symmApply,
+  simp only [snf, Basis.map_apply, Submodule.comapSubtypeEquivOfLe_symm_apply,
     Submodule.coe_smul_of_tower, RelEmbedding.coeFn_toEmbedding]
 #align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLe
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

A PR analogous to #12338 and #12361: reformatting proofs following the multiple goals linter of #12339.

@@ -124,23 +124,23 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
   refine' dvd_trans _ d_dvd_right
   rw [dvd_generator_iff, Ideal.span, ←
     span_singleton_generator (Submodule.span R {a, ψ ⟨y, hNO yN⟩})]
-  obtain ⟨r₁, r₂, d_eq⟩ : ∃ r₁ r₂ : R, d = r₁ * a + r₂ * ψ ⟨y, hNO yN⟩ := by
-    obtain ⟨r₁, r₂', hr₂', hr₁⟩ :=
-      mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, ψ ⟨y, hNO yN⟩}))
-    obtain ⟨r₂, rfl⟩ := mem_span_singleton.mp hr₂'
-    exact ⟨r₁, r₂, hr₁⟩
-  let ψ' : O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
-  have : span R {d} ≤ ψ'.submoduleImage N := by
-    rw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]
-    refine' ⟨y, yN, _⟩
-    change r₁ * ϕ ⟨y, hNO yN⟩ + r₂ * ψ ⟨y, hNO yN⟩ = d
-    rw [d_eq, ϕy_eq]
-  refine'
-    le_antisymm (this.trans (le_of_eq _)) (Ideal.span_singleton_le_span_singleton.mpr d_dvd_left)
-  rw [span_singleton_generator]
-  apply (le_trans _ this).eq_of_not_gt (hϕ ψ')
-  rw [← span_singleton_generator (ϕ.submoduleImage N)]
-  exact Ideal.span_singleton_le_span_singleton.mpr d_dvd_left
+  · obtain ⟨r₁, r₂, d_eq⟩ : ∃ r₁ r₂ : R, d = r₁ * a + r₂ * ψ ⟨y, hNO yN⟩ := by
+      obtain ⟨r₁, r₂', hr₂', hr₁⟩ :=
+        mem_span_insert.mp (IsPrincipal.generator_mem (Submodule.span R {a, ψ ⟨y, hNO yN⟩}))
+      obtain ⟨r₂, rfl⟩ := mem_span_singleton.mp hr₂'
+      exact ⟨r₁, r₂, hr₁⟩
+    let ψ' : O →ₗ[R] R := r₁ • ϕ + r₂ • ψ
+    have : span R {d} ≤ ψ'.submoduleImage N := by
+      rw [span_le, singleton_subset_iff, SetLike.mem_coe, LinearMap.mem_submoduleImage_of_le hNO]
+      refine' ⟨y, yN, _⟩
+      change r₁ * ϕ ⟨y, hNO yN⟩ + r₂ * ψ ⟨y, hNO yN⟩ = d
+      rw [d_eq, ϕy_eq]
+    refine'
+      le_antisymm (this.trans (le_of_eq _)) (Ideal.span_singleton_le_span_singleton.mpr d_dvd_left)
+    rw [span_singleton_generator]
+    apply (le_trans _ this).eq_of_not_gt (hϕ ψ')
+    rw [← span_singleton_generator (ϕ.submoduleImage N)]
+    exact Ideal.span_singleton_le_span_singleton.mpr d_dvd_left
   · exact subset_span (mem_insert _ _)
 #align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvd
 
@@ -291,15 +291,15 @@ theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R
   cases nonempty_fintype ι
   induction' N using inductionOnRank with N ih
   · exact b
-  let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
-  by_cases N_bot : N = ⊥
-  · subst N_bot
-    exact ⟨0, ⟨Basis.empty _⟩⟩
-  obtain ⟨y, -, a, hay, M', -, N', N'_le_N, -, -, ay_ortho, h'⟩ :=
-    Submodule.basis_of_pid_aux ⊤ N b' N_bot le_top
-  obtain ⟨n', ⟨bN'⟩⟩ := ih N' N'_le_N _ hay ay_ortho
-  obtain ⟨bN, _hbN⟩ := h' n' bN'
-  exact ⟨n' + 1, ⟨bN⟩⟩
+  · let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
+    by_cases N_bot : N = ⊥
+    · subst N_bot
+      exact ⟨0, ⟨Basis.empty _⟩⟩
+    obtain ⟨y, -, a, hay, M', -, N', N'_le_N, -, -, ay_ortho, h'⟩ :=
+      Submodule.basis_of_pid_aux ⊤ N b' N_bot le_top
+    obtain ⟨n', ⟨bN'⟩⟩ := ih N' N'_le_N _ hay ay_ortho
+    obtain ⟨bN, _hbN⟩ := h' n' bN'
+    exact ⟨n' + 1, ⟨bN⟩⟩
   infer_instance
 #align submodule.nonempty_basis_of_pid Submodule.nonempty_basis_of_pid
 
@@ -513,19 +513,19 @@ theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M)
   revert N
   induction' O using inductionOnRank with M0 ih
   · exact b
-  intro N N_le_M0
-  obtain ⟨m, b'M⟩ := M0.basisOfPid b
-  by_cases N_bot : N = ⊥
-  · subst N_bot
-    exact ⟨0, m, Nat.zero_le _, b'M, Basis.empty _, finZeroElim, finZeroElim⟩
-  obtain ⟨y, hy, a, _, M', M'_le_M, N', _, N'_le_M', y_ortho, _, h⟩ :=
-    Submodule.basis_of_pid_aux M0 N b'M N_bot N_le_M0
-
-  obtain ⟨n', m', hn'm', bM', bN', as', has'⟩ := ih M' M'_le_M y hy y_ortho N' N'_le_M'
-  obtain ⟨bN, h'⟩ := h n' bN'
-  obtain ⟨hmn, bM, h''⟩ := h' m' hn'm' bM'
-  obtain ⟨as, has⟩ := h'' as' has'
-  exact ⟨_, _, hmn, bM, bN, as, has⟩
+  · intro N N_le_M0
+    obtain ⟨m, b'M⟩ := M0.basisOfPid b
+    by_cases N_bot : N = ⊥
+    · subst N_bot
+      exact ⟨0, m, Nat.zero_le _, b'M, Basis.empty _, finZeroElim, finZeroElim⟩
+    obtain ⟨y, hy, a, _, M', M'_le_M, N', _, N'_le_M', y_ortho, _, h⟩ :=
+      Submodule.basis_of_pid_aux M0 N b'M N_bot N_le_M0
+
+    obtain ⟨n', m', hn'm', bM', bN', as', has'⟩ := ih M' M'_le_M y hy y_ortho N' N'_le_M'
+    obtain ⟨bN, h'⟩ := h n' bN'
+    obtain ⟨hmn, bM, h''⟩ := h' m' hn'm' bM'
+    obtain ⟨as, has⟩ := h'' as' has'
+    exact ⟨_, _, hmn, bM, bN, as, has⟩
 -- Porting note: Lean generates a goal Fintype ι for some reason
   infer_instance
 #align submodule.exists_smith_normal_form_of_le Submodule.exists_smith_normal_form_of_le

Reference the newly created issues #12094 and #12096, as well as the pre-existing #5171. Change all references to #10927 to #5171. Some of these changes were not labelled as "porting note"; change this for good measure.

@@ -426,7 +426,7 @@ section SmithNormal
 /-- A Smith normal form basis for a submodule `N` of a module `M` consists of
 bases for `M` and `N` such that the inclusion map `N → M` can be written as a
 (rectangular) matrix with `a` along the diagonal: in Smith normal form. -/
--- Porting note: @[nolint has_nonempty_instance]
+-- Porting note(#5171): @[nolint has_nonempty_instance]
 structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type*) (n : ℕ) where
   /-- The basis of M. -/
   bM : Basis ι R M

Empty lines were removed by executing the following Python script twice

import os
import re


# Loop through each file in the repository
for dir_path, dirs, files in os.walk('.'):
  for filename in files:
    if filename.endswith('.lean'):
      file_path = os.path.join(dir_path, filename)

      # Open the file and read its contents
      with open(file_path, 'r') as file:
        content = file.read()

      # Use a regular expression to replace sequences of "variable" lines separated by empty lines
      # with sequences without empty lines
      modified_content = re.sub(r'(variable.*\n)\n(variable(?! .* in))', r'\1\2', content)

      # Write the modified content back to the file
      with open(file_path, 'w') as file:
        file.write(modified_content)

@@ -54,7 +54,6 @@ universe u v
 section Ring
 
 variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
-
 variable {ι : Type*} (b : Basis ι R M)
 
 open Submodule.IsPrincipal Submodule
@@ -89,7 +88,6 @@ end Ring
 section IsDomain
 
 variable {ι : Type*} {R : Type*} [CommRing R] [IsDomain R]
-
 variable {M : Type*} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal Set Submodule
@@ -109,7 +107,6 @@ section PrincipalIdealDomain
 open Submodule.IsPrincipal Set Submodule
 
 variable {ι : Type*} {R : Type*} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
-
 variable {M : Type*} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

• converts "--porting note" into "-- Porting note";
• converts "porting note" into "Porting note".

@@ -681,7 +681,7 @@ theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   simp [hi]
 #align ideal.smith_coeffs_ne_zero Ideal.smithCoeffs_ne_zero
 
--- porting note: can be inferred in Lean 4 so no longer necessary
+-- Porting note: can be inferred in Lean 4 so no longer necessary
 #noalign has_quotient.quotient.module
 
 end Ideal

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -228,7 +228,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Mod
   have M'_le_M : M' ≤ M := M.map_subtype_le (LinearMap.ker ϕ)
   have N'_le_M' : N' ≤ M' := by
     intro x hx
-    simp only [mem_map, LinearMap.mem_ker] at hx ⊢
+    simp only [N', mem_map, LinearMap.mem_ker] at hx ⊢
     obtain ⟨⟨x, xN⟩, hx, rfl⟩ := hx
     exact ⟨⟨x, N_le_M xN⟩, hx, rfl⟩
   have N'_le_N : N' ≤ N := N.map_subtype_le (LinearMap.ker (ϕ.comp (inclusion N_le_M)))
@@ -568,7 +568,7 @@ noncomputable def Submodule.smithNormalForm [Finite ι] (b : Basis ι R M) (N :
   let bM' := bM.map (LinearEquiv.ofTop _ rfl)
   let e := bM'.indexEquiv b
   ⟨n, bM'.reindex e, bN.map (comapSubtypeEquivOfLe le_top), f.trans e.toEmbedding, a, fun i ↦ by
-    simp only [snf, Basis.map_apply, LinearEquiv.ofTop_apply, Submodule.coe_smul_of_tower,
+    simp only [bM', snf, Basis.map_apply, LinearEquiv.ofTop_apply, Submodule.coe_smul_of_tower,
       Submodule.comapSubtypeEquivOfLe_apply_coe, Basis.reindex_apply,
       Equiv.toEmbedding_apply, Function.Embedding.trans_apply, Equiv.symm_apply_apply]⟩
 #align submodule.smith_normal_form Submodule.smithNormalForm

This uses the improved shake script from #9772 to reduce imports across mathlib. The corresponding noshake.json file has been added to #9772.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 -/
 import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
-import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
+import Mathlib.LinearAlgebra.FreeModule.Basic
 
 #align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
 

Add also a NoZeroSMulDivisors instance.

These instances, in particular, imply directly that integral ideals of number fields are free and finite $\mathbb{Z}$-modules.

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 -/
 import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
-import Mathlib.LinearAlgebra.FreeModule.Basic
+import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
 
 #align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
 
@@ -413,6 +413,13 @@ instance Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMu
   exact Module.Free.of_basis b
 #align module.free_of_finite_type_torsion_free' Module.free_of_finite_type_torsion_free'
 
+instance {S : Type*} [CommRing S] [Algebra R S] {I : Ideal S} [hI₁ : Module.Finite R I]
+    [hI₂ : NoZeroSMulDivisors R I] : Module.Free R I := by
+  have : Module.Finite R (restrictScalars R I) := hI₁
+  have : NoZeroSMulDivisors R (restrictScalars R I) := hI₂
+  change Module.Free R (restrictScalars R I)
+  exact Module.free_of_finite_type_torsion_free'
+
 theorem Module.free_iff_noZeroSMulDivisors [Module.Finite R M] :
     Module.Free R M ↔ NoZeroSMulDivisors R M :=
   ⟨fun _ ↦ inferInstance, fun _ ↦ inferInstance⟩

This is a follow-up to #9215. It changes the following theorems and definitions:

• IsOpen.exists_subset_affineIndependent_span_eq_top
• IsConformalMap
• SimpleGraph.induce_connected_of_patches
• Submonoid.exists_list_of_mem_closure
• AddSubmonoid.exists_list_of_mem_closure
• AffineSubspace.mem_affineSpan_insert_iff
• AffineBasis.exists_affine_subbasis
• exists_affineIndependent
• LinearMap.mem_submoduleImage
• Basis.basis_singleton_iff
• atom_iff_nonzero_span
• finrank_eq_one_iff'
• Submodule.basis_of_pid_aux
• exists_linearIndependent_extension
• exists_linearIndependent
• countable_cover_nhdsWithin_of_sigma_compact
• mem_residual

Also deprecate ENNReal.exists_ne_top'.

@@ -160,21 +160,17 @@ but must also feed in a basis for `M` using `basis_of_pid` to keep the induction
 -/
 theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
     (M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
-    ∃ y ∈ M,
-      ∃ (a : R) (_ : a • y ∈ N),
-        ∃ M' ≤ M,
-          ∃ N' ≤ N,
-            ∃ (_N'_le_M' : N' ≤ M') (_y_ortho_M' :
-              ∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) (_ay_ortho_N' :
-              ∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0),
-              ∀ (n') (bN' : Basis (Fin n') R N'),
-                ∃ bN : Basis (Fin (n' + 1)) R N,
-                  ∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
-                    ∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
-                      ∀ (as : Fin n' → R)
-                        (_h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)),
-                        ∃ as' : Fin (n' + 1) → R,
-                          ∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O) := by
+    ∃ y ∈ M, ∃ a : R, a • y ∈ N ∧ ∃ M' ≤ M, ∃ N' ≤ N,
+      N' ≤ M' ∧ (∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) ∧
+      (∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0) ∧
+      ∀ (n') (bN' : Basis (Fin n') R N'),
+        ∃ bN : Basis (Fin (n' + 1)) R N,
+          ∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
+            ∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
+              ∀ as : Fin n' → R,
+                (∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)) →
+                  ∃ as' : Fin (n' + 1) → R,
+                    ∀ i : Fin (n' + 1), (bN i : O) = as' i • (bM (Fin.castLE hnm i) : O) := by
   -- Let `ϕ` be a maximal projection of `M` onto `R`, in the sense that there is
   -- no `ψ` whose image of `N` is larger than `ϕ`'s image of `N`.
   have : ∃ ϕ : M →ₗ[R] R, ∀ ψ : M →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N := by

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -411,14 +411,16 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree' [Module.Finite R M]
   Module.basisOfFiniteTypeTorsionFree Module.Finite.exists_fin.choose_spec.choose_spec
 #align module.basis_of_finite_type_torsion_free' Module.basisOfFiniteTypeTorsionFree'
 
--- It would be nice to make this an instance but it is empirically problematic, possibly because
--- of the loop that it causes with `Module.Free.noZeroSMulDivisors`
-theorem Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMulDivisors R M] :
+instance Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMulDivisors R M] :
     Module.Free R M := by
   obtain ⟨n, b⟩ : Σn, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree'
   exact Module.Free.of_basis b
 #align module.free_of_finite_type_torsion_free' Module.free_of_finite_type_torsion_free'
 
+theorem Module.free_iff_noZeroSMulDivisors [Module.Finite R M] :
+    Module.Free R M ↔ NoZeroSMulDivisors R M :=
+  ⟨fun _ ↦ inferInstance, fun _ ↦ inferInstance⟩
+
 section SmithNormal
 
 /-- A Smith normal form basis for a submodule `N` of a module `M` consists of

The files Mathlib.LinearAlgebra.FreeModule.Rank, Mathlib.LinearAlgebra.FreeModule.Finite.Rank, Mathlib.LinearAlgebra.Dimension and Mathlib.LinearAlgebra.Finrank were reorganized into a folder Mathlib.LinearAlgebra.Dimension, containing the following files

• Basic.lean: Contains the definition of Module.rank.
• Finrank.lean: Contains the definition of FiniteDimensional.finrank.
• StrongRankCondition.lean: Contains results about rank and finrank over rings satisfying strong rank condition
• Free.lean: Contains results about rank and finrank of free modules
• Finite.lean: Contains conditions or consequences for rank to be finite or zero
• Constructions.lean: Contains the calculation of the rank of various constructions.
• DivisionRing.lean: Contains results about rank and finrank of spaces over division rings.
• LinearMap.lean: Contains results about LinearMap.rank

API changes: IsNoetherian.rank_lt_aleph0 and FiniteDimensional.rank_lt_aleph0 are replaced with rank_lt_aleph0. Module.Free.finite_basis was renamed to Module.Finite.finite_basis. FiniteDimensional.finrank_eq_rank was renamed to finrank_eq_rank. rank_eq_cardinal_basis and rank_eq_cardinal_basis' were removed in favour of Basis.mk_eq_mk and Basis.mk_eq_mk''.

Co-authored-by: Andrew Yang <36414270+erdOne@users.noreply.github.com>

@@ -3,11 +3,8 @@ Copyright (c) 2020 Anne Baanen. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 -/
-import Mathlib.LinearAlgebra.Dimension
+import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
 import Mathlib.LinearAlgebra.FreeModule.Basic
-import Mathlib.LinearAlgebra.Matrix.ToLin
-import Mathlib.RingTheory.PrincipalIdealDomain
-import Mathlib.RingTheory.Finiteness
 
 #align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
 

Search for [∀∃].*(_ and manually replace some occurrences with more readable versions. In case of ∀, the new expressions are defeq to the old ones. In case of ∃, they differ by exists_prop.

In some rare cases, golf proofs that needed fixing.

@@ -359,7 +359,7 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M
     let I : Set ι := this.choose
     obtain
       ⟨indepI : LinearIndependent R (s ∘ (fun x => x) : I → M), hI :
-        ∀ (i) (_ : i ∉ I), ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I)⟩ :=
+        ∀ i ∉ I, ∃ a : R, a ≠ 0 ∧ a • s i ∈ span R (s '' I)⟩ :=
       this.choose_spec
     let N := span R (range <| (s ∘ (fun x => x) : I → M))
     -- same as `span R (s '' I)` but more convenient

This matches Set.inclusion, Subring.inclusion, Subalgebra.inclusion, etc.

Also renames the homOfLe spellings in Algebra/Lie to match.

Note that we leave LieSubalgebra.ofLe, as this is a completely different statement!

As requested by @alreadydone.

@@ -231,14 +231,14 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Mod
   have ϕy'_ne_zero : ϕ ⟨y', y'M⟩ ≠ 0 := by simpa only [ϕy'_eq] using one_ne_zero
   -- `M' := ker (ϕ : M → R)` is smaller than `M` and `N' := ker (ϕ : N → R)` is smaller than `N`.
   let M' : Submodule R O := ϕ.ker.map M.subtype
-  let N' : Submodule R O := (ϕ.comp (ofLe N_le_M)).ker.map N.subtype
+  let N' : Submodule R O := (ϕ.comp (inclusion N_le_M)).ker.map N.subtype
   have M'_le_M : M' ≤ M := M.map_subtype_le (LinearMap.ker ϕ)
   have N'_le_M' : N' ≤ M' := by
     intro x hx
     simp only [mem_map, LinearMap.mem_ker] at hx ⊢
     obtain ⟨⟨x, xN⟩, hx, rfl⟩ := hx
     exact ⟨⟨x, N_le_M xN⟩, hx, rfl⟩
-  have N'_le_N : N' ≤ N := N.map_subtype_le (LinearMap.ker (ϕ.comp (ofLe N_le_M)))
+  have N'_le_N : N' ≤ N := N.map_subtype_le (LinearMap.ker (ϕ.comp (inclusion N_le_M)))
   -- So fill in those results as well.
   refine' ⟨M', M'_le_M, N', N'_le_N, N'_le_M', _⟩
   -- Note that `y'` is orthogonal to `M'`.
@@ -284,7 +284,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Mod
   · simp only [Fin.cons_zero, Fin.castLE_zero]
     exact a_smul_y'.symm
   · rw [Fin.castLE_succ]
-    simp only [Fin.cons_succ, Function.comp_apply, coe_ofLe, map_coe, coeSubtype, h i]
+    simp only [Fin.cons_succ, Function.comp_apply, coe_inclusion, map_coe, coeSubtype, h i]
 #align submodule.basis_of_pid_aux Submodule.basis_of_pid_aux
 
 /-- A submodule of a free `R`-module of finite rank is also a free `R`-module of finite rank,

And fix some names in comments where this revealed issues

@@ -33,7 +33,7 @@ In this section, `M` is a free and finitely generated `R`-module, and
   `P N` follows from `P N'` for all `N'` that are of lower rank, then `P` holds
    on all submodules
 
- - `submodule.exists_basis_of_pid`: if `R` is a PID, then `N : Submodule R M` is
+ - `Submodule.exists_basis_of_pid`: if `R` is a PID, then `N : Submodule R M` is
    free and finitely generated. This is the first part of the structure theorem
    for modules.
 

@@ -450,7 +450,7 @@ lemma repr_eq_zero_of_nmem_range {i : ι} (hi : i ∉ Set.range snf.f) :
   obtain ⟨m, hm⟩ := m
   obtain ⟨c, rfl⟩ := snf.bN.mem_submodule_iff.mp hm
   replace hi : ∀ j, snf.f j ≠ i := by simpa using hi
-  simp [Finsupp.single_apply, hi, snf.snf]
+  simp [Finsupp.single_apply, hi, snf.snf, map_finsupp_sum]
 
 lemma le_ker_coord_of_nmem_range {i : ι} (hi : i ∉ Set.range snf.f) :
     N ≤ LinearMap.ker (snf.bM.coord i) :=
@@ -461,9 +461,10 @@ lemma le_ker_coord_of_nmem_range {i : ι} (hi : i ∉ Set.range snf.f) :
   obtain ⟨m, hm⟩ := m
   obtain ⟨c, rfl⟩ := snf.bN.mem_submodule_iff.mp hm
   replace hm : (⟨Finsupp.sum c fun i t ↦ t • (↑(snf.bN i) : M), hm⟩ : N) =
-      Finsupp.sum c fun i t ↦ t • ⟨snf.bN i, (snf.bN i).2⟩ := by ext; change _ = N.subtype _; simp
+      Finsupp.sum c fun i t ↦ t • ⟨snf.bN i, (snf.bN i).2⟩ := by
+    ext; change _ = N.subtype _; simp [map_finsupp_sum]
   classical
-  simp_rw [hm, map_smul, LinearEquiv.map_finsupp_sum, map_smul, Subtype.coe_eta, repr_self,
+  simp_rw [hm, map_smul, map_finsupp_sum, map_smul, Subtype.coe_eta, repr_self,
     Finsupp.smul_single, smul_eq_mul, mul_one, Finsupp.sum_single, Finsupp.smul_apply, snf.snf,
     map_smul, repr_self, Finsupp.smul_single, smul_eq_mul, mul_one, Finsupp.sum_apply,
     Finsupp.single_apply, EmbeddingLike.apply_eq_iff_eq, Finsupp.sum_ite_eq',

Also _root_.map_smul when in the neighbourhood.

@@ -210,7 +210,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Mod
   have mk_y' : (⟨y', y'M⟩ : M) = ∑ i, c i • b'M i :=
     Subtype.ext
       (show y' = M.subtype _ by
-        simp only [LinearMap.map_sum, LinearMap.map_smul]
+        simp only [map_sum, map_smul]
         rfl)
   have a_smul_y' : a • y' = y := by
     refine Subtype.mk_eq_mk.mp (show (a • ⟨y', y'M⟩ : M) = ⟨y, N_le_M yN⟩ from ?_)

@@ -5,6 +5,7 @@ Authors: Anne Baanen
 -/
 import Mathlib.LinearAlgebra.Dimension
 import Mathlib.LinearAlgebra.FreeModule.Basic
+import Mathlib.LinearAlgebra.Matrix.ToLin
 import Mathlib.RingTheory.PrincipalIdealDomain
 import Mathlib.RingTheory.Finiteness
 
@@ -440,6 +441,61 @@ structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type*) (n : ℕ) where
   snf : ∀ i, (bN i : M) = a i • bM (f i)
 #align basis.smith_normal_form Basis.SmithNormalForm
 
+namespace Basis.SmithNormalForm
+
+variable {n : ℕ} {N : Submodule R M} (snf : Basis.SmithNormalForm N ι n) (m : N)
+
+lemma repr_eq_zero_of_nmem_range {i : ι} (hi : i ∉ Set.range snf.f) :
+    snf.bM.repr m i = 0 := by
+  obtain ⟨m, hm⟩ := m
+  obtain ⟨c, rfl⟩ := snf.bN.mem_submodule_iff.mp hm
+  replace hi : ∀ j, snf.f j ≠ i := by simpa using hi
+  simp [Finsupp.single_apply, hi, snf.snf]
+
+lemma le_ker_coord_of_nmem_range {i : ι} (hi : i ∉ Set.range snf.f) :
+    N ≤ LinearMap.ker (snf.bM.coord i) :=
+  fun m hm ↦ snf.repr_eq_zero_of_nmem_range ⟨m, hm⟩ hi
+
+@[simp] lemma repr_apply_embedding_eq_repr_smul {i : Fin n} :
+    snf.bM.repr m (snf.f i) = snf.bN.repr (snf.a i • m) i := by
+  obtain ⟨m, hm⟩ := m
+  obtain ⟨c, rfl⟩ := snf.bN.mem_submodule_iff.mp hm
+  replace hm : (⟨Finsupp.sum c fun i t ↦ t • (↑(snf.bN i) : M), hm⟩ : N) =
+      Finsupp.sum c fun i t ↦ t • ⟨snf.bN i, (snf.bN i).2⟩ := by ext; change _ = N.subtype _; simp
+  classical
+  simp_rw [hm, map_smul, LinearEquiv.map_finsupp_sum, map_smul, Subtype.coe_eta, repr_self,
+    Finsupp.smul_single, smul_eq_mul, mul_one, Finsupp.sum_single, Finsupp.smul_apply, snf.snf,
+    map_smul, repr_self, Finsupp.smul_single, smul_eq_mul, mul_one, Finsupp.sum_apply,
+    Finsupp.single_apply, EmbeddingLike.apply_eq_iff_eq, Finsupp.sum_ite_eq',
+    Finsupp.mem_support_iff, ite_not, mul_comm, ite_eq_right_iff]
+  exact fun a ↦ (mul_eq_zero_of_right _ a).symm
+
+@[simp] lemma repr_comp_embedding_eq_smul :
+    snf.bM.repr m ∘ snf.f = snf.a • (snf.bN.repr m : Fin n → R) := by
+  ext i
+  simp [Pi.smul_apply (snf.a i)]
+
+@[simp] lemma coord_apply_embedding_eq_smul_coord {i : Fin n} :
+    snf.bM.coord (snf.f i) ∘ₗ N.subtype = snf.a i • snf.bN.coord i := by
+  ext m
+  simp [Pi.smul_apply (snf.a i)]
+
+/-- Given a Smith-normal-form pair of bases for `N ⊆ M`, and a linear endomorphism `f` of `M`
+that preserves `N`, the diagonal of the matrix of the restriction `f` to `N` does not depend on
+which of the two bases for `N` is used. -/
+@[simp]
+lemma toMatrix_restrict_eq_toMatrix [Fintype ι] [DecidableEq ι]
+    (f : M →ₗ[R] M) (hf : ∀ x, f x ∈ N) (hf' : ∀ x ∈ N, f x ∈ N := fun x _ ↦ hf x) {i : Fin n} :
+    LinearMap.toMatrix snf.bN snf.bN (LinearMap.restrict f hf') i i =
+    LinearMap.toMatrix snf.bM snf.bM f (snf.f i) (snf.f i) := by
+  rw [LinearMap.toMatrix_apply, LinearMap.toMatrix_apply,
+    snf.repr_apply_embedding_eq_repr_smul ⟨_, (hf _)⟩]
+  congr
+  ext
+  simp [snf.snf]
+
+end Basis.SmithNormalForm
+
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free
 and we can find a basis for `M` and `N` such that the inclusion map is a diagonal matrix
 in Smith normal form.

@@ -413,6 +413,8 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree' [Module.Finite R M]
   Module.basisOfFiniteTypeTorsionFree Module.Finite.exists_fin.choose_spec.choose_spec
 #align module.basis_of_finite_type_torsion_free' Module.basisOfFiniteTypeTorsionFree'
 
+-- It would be nice to make this an instance but it is empirically problematic, possibly because
+-- of the loop that it causes with `Module.Free.noZeroSMulDivisors`
 theorem Module.free_of_finite_type_torsion_free' [Module.Finite R M] [NoZeroSMulDivisors R M] :
     Module.Free R M := by
   obtain ⟨n, b⟩ : Σn, Basis (Fin n) R M := Module.basisOfFiniteTypeTorsionFree'

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -57,7 +57,7 @@ section Ring
 
 variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup M] [Module R M]
 
-variable {ι : Type _} (b : Basis ι R M)
+variable {ι : Type*} (b : Basis ι R M)
 
 open Submodule.IsPrincipal Submodule
 
@@ -90,9 +90,9 @@ end Ring
 
 section IsDomain
 
-variable {ι : Type _} {R : Type _} [CommRing R] [IsDomain R]
+variable {ι : Type*} {R : Type*} [CommRing R] [IsDomain R]
 
-variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
+variable {M : Type*} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal Set Submodule
 
@@ -110,9 +110,9 @@ section PrincipalIdealDomain
 
 open Submodule.IsPrincipal Set Submodule
 
-variable {ι : Type _} {R : Type _} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
+variable {ι : Type*} {R : Type*} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
 
-variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
+variable {M : Type*} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal
 
@@ -160,7 +160,7 @@ For `basis_of_pid` we only need the first half and can fix `M = ⊤`,
 for `smith_normal_form` we need the full statement,
 but must also feed in a basis for `M` using `basis_of_pid` to keep the induction going.
 -/
-theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Module R O]
+theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type*} [AddCommGroup O] [Module R O]
     (M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
     ∃ y ∈ M,
       ∃ (a : R) (_ : a • y ∈ N),
@@ -294,7 +294,7 @@ see `Submodule.basisOfPid`.
 
 See also the stronger version `Submodule.smithNormalForm`.
 -/
-theorem Submodule.nonempty_basis_of_pid {ι : Type _} [Finite ι] (b : Basis ι R M)
+theorem Submodule.nonempty_basis_of_pid {ι : Type*} [Finite ι] (b : Basis ι R M)
     (N : Submodule R M) : ∃ n : ℕ, Nonempty (Basis (Fin n) R N) := by
   haveI := Classical.decEq M
   cases nonempty_fintype ι
@@ -317,12 +317,12 @@ if `R` is a principal ideal domain.
 
 See also the stronger version `Submodule.smithNormalForm`.
 -/
-noncomputable def Submodule.basisOfPid {ι : Type _} [Finite ι] (b : Basis ι R M)
+noncomputable def Submodule.basisOfPid {ι : Type*} [Finite ι] (b : Basis ι R M)
     (N : Submodule R M) : Σn : ℕ, Basis (Fin n) R N :=
   ⟨_, (N.nonempty_basis_of_pid b).choose_spec.some⟩
 #align submodule.basis_of_pid Submodule.basisOfPid
 
-theorem Submodule.basisOfPid_bot {ι : Type _} [Finite ι] (b : Basis ι R M) :
+theorem Submodule.basisOfPid_bot {ι : Type*} [Finite ι] (b : Basis ι R M) :
     Submodule.basisOfPid b ⊥ = ⟨0, Basis.empty _⟩ := by
   obtain ⟨n, b'⟩ := Submodule.basisOfPid b ⊥
   let e : Fin n ≃ Fin 0 := b'.indexEquiv (Basis.empty _ : Basis (Fin 0) R (⊥ : Submodule R M))
@@ -335,7 +335,7 @@ if `R` is a principal ideal domain.
 
 See also the stronger version `Submodule.smithNormalFormOfLE`.
 -/
-noncomputable def Submodule.basisOfPidOfLE {ι : Type _} [Finite ι] {N O : Submodule R M}
+noncomputable def Submodule.basisOfPidOfLE {ι : Type*} [Finite ι] {N O : Submodule R M}
     (hNO : N ≤ O) (b : Basis ι R O) : Σn : ℕ, Basis (Fin n) R N :=
   let ⟨n, bN'⟩ := Submodule.basisOfPid b (N.comap O.subtype)
   ⟨n, bN'.map (Submodule.comapSubtypeEquivOfLe hNO)⟩
@@ -343,7 +343,7 @@ noncomputable def Submodule.basisOfPidOfLE {ι : Type _} [Finite ι] {N O : Subm
 
 /-- A submodule inside the span of a linear independent family is a free `R`-module of finite rank,
 if `R` is a principal ideal domain. -/
-noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι → M}
+noncomputable def Submodule.basisOfPidOfLESpan {ι : Type*} [Finite ι] {b : ι → M}
     (hb : LinearIndependent R b) {N : Submodule R M} (le : N ≤ Submodule.span R (Set.range b)) :
     Σn : ℕ, Basis (Fin n) R N :=
   Submodule.basisOfPidOfLE le (Basis.span hb)
@@ -425,7 +425,7 @@ section SmithNormal
 bases for `M` and `N` such that the inclusion map `N → M` can be written as a
 (rectangular) matrix with `a` along the diagonal: in Smith normal form. -/
 -- Porting note: @[nolint has_nonempty_instance]
-structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type _) (n : ℕ) where
+structure Basis.SmithNormalForm (N : Submodule R M) (ι : Type*) (n : ℕ) where
   /-- The basis of M. -/
   bM : Basis ι R M
   /-- The basis of N. -/
@@ -514,7 +514,7 @@ noncomputable def Submodule.smithNormalForm [Finite ι] (b : Basis ι R M) (N :
 
 section Ideal
 
-variable {S : Type _} [CommRing S] [IsDomain S] [Algebra R S]
+variable {S : Type*} [CommRing S] [IsDomain S] [Algebra R S]
 
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
@@ -631,7 +631,7 @@ end PrincipalIdealDomain
 
 /-- A set of linearly independent vectors in a module `M` over a semiring `S` is also linearly
 independent over a subring `R` of `K`. -/
-theorem LinearIndependent.restrict_scalars_algebras {R S M ι : Type _} [CommSemiring R] [Semiring S]
+theorem LinearIndependent.restrict_scalars_algebras {R S M ι : Type*} [CommSemiring R] [Semiring S]
     [AddCommMonoid M] [Algebra R S] [Module R M] [Module S M] [IsScalarTower R S M]
     (hinj : Function.Injective (algebraMap R S)) {v : ι → M} (li : LinearIndependent S v) :
     LinearIndependent R v :=

@@ -255,7 +255,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
     rw [mul_comm, mul_smul, hc]
   -- So we can extend a basis for `N'` with `y`
   refine' ⟨y'_ortho_M', ay'_ortho_N', fun n' bN' ↦ ⟨_, _⟩⟩
-  · refine' Basis.mkFinConsOfLe y yN bN' N'_le_N _ _
+  · refine' Basis.mkFinConsOfLE y yN bN' N'_le_N _ _
     · intro c z zN' hc
       refine' ay'_ortho_N' c z zN' _
       rwa [← a_smul_y'] at hc
@@ -268,7 +268,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   -- And extend a basis for `M'` with `y'`
   intro m' hn'm' bM'
   refine' ⟨Nat.succ_le_succ hn'm', _, _⟩
-  · refine' Basis.mkFinConsOfLe y' y'M bM' M'_le_M y'_ortho_M' _
+  · refine' Basis.mkFinConsOfLE y' y'M bM' M'_le_M y'_ortho_M' _
     intro z zM
     refine' ⟨-ϕ ⟨z, zM⟩, ⟨⟨z, zM⟩ - ϕ ⟨z, zM⟩ • ⟨y', y'M⟩, LinearMap.mem_ker.mpr _, _⟩⟩
     · rw [LinearMap.map_sub, LinearMap.map_smul, ϕy'_eq, smul_eq_mul, mul_one, sub_self]
@@ -278,7 +278,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   intro as h
   refine' ⟨Fin.cons a as, _⟩
   intro i
-  rw [Basis.coe_mkFinConsOfLe, Basis.coe_mkFinConsOfLe]
+  rw [Basis.coe_mkFinConsOfLE, Basis.coe_mkFinConsOfLE]
   refine' Fin.cases _ (fun i ↦ _) i
   · simp only [Fin.cons_zero, Fin.castLE_zero]
     exact a_smul_y'.symm

Add the following result:

theorem mem_span_integralBasis {x : K} :
    x ∈ Submodule.span ℤ (Set.range (integralBasis K)) ↔ x ∈ 𝓞 K

that is, integralBasis is indeed a ℤ-basis of the ring of integers.

Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com>

Co-authored-by: Riccardo Brasca <riccardo.brasca@gmail.com>

@@ -385,7 +385,7 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M
     have : LinearMap.range φ ≤ N := by
       -- as announced, `A • M ⊆ N`
       suffices ∀ i, φ (s i) ∈ N by
-        rw [LinearMap.range_eq_map, ← hs, φ.map_span_le]
+        rw [LinearMap.range_eq_map, ← hs, map_span_le]
         rintro _ ⟨i, rfl⟩
         apply this
       intro i

@@ -188,7 +188,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
     exact ⟨ϕ, fun ψ hψ ↦ P_max _ ⟨_, rfl⟩ hψ⟩
   let ϕ := this.choose
   have ϕ_max := this.choose_spec
-  -- Since `ϕ(N)` is a `R`-submodule of the PID `R`,
+  -- Since `ϕ(N)` is an `R`-submodule of the PID `R`,
   -- it is principal and generated by some `a`.
   let a := generator (ϕ.submoduleImage N)
   have a_mem : a ∈ ϕ.submoduleImage N := generator_mem _

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,17 +2,14 @@
 Copyright (c) 2020 Anne Baanen. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
-
-! This file was ported from Lean 3 source module linear_algebra.free_module.pid
-! leanprover-community/mathlib commit d87199d51218d36a0a42c66c82d147b5a7ff87b3
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.LinearAlgebra.Dimension
 import Mathlib.LinearAlgebra.FreeModule.Basic
 import Mathlib.RingTheory.PrincipalIdealDomain
 import Mathlib.RingTheory.Finiteness
 
+#align_import linear_algebra.free_module.pid from "leanprover-community/mathlib"@"d87199d51218d36a0a42c66c82d147b5a7ff87b3"
+
 /-! # Free modules over PID
 
 A free `R`-module `M` is a module with a basis over `R`,

• depends on: https://github.com/leanprover/std4/pull/163
• depends on: #5584
• depends on: #5715
• depends on: #5729
• depends on: https://github.com/leanprover/std4/pull/173

Co-authored-by: Komyyy <pol_tta@outlook.jp> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -488,11 +488,11 @@ noncomputable def Submodule.smithNormalFormOfLE [Finite ι] (b : Basis ι R M) (
     (N_le_O : N ≤ O) : Σo n : ℕ, Basis.SmithNormalForm (N.comap O.subtype) (Fin o) n := by
   choose n o hno bO bN a snf using N.exists_smith_normal_form_of_le b O N_le_O
   refine'
-    ⟨o, n, bO, bN.map (comapSubtypeEquivOfLe N_le_O).symm, (Fin.castLE hno).toEmbedding, a,
+    ⟨o, n, bO, bN.map (comapSubtypeEquivOfLe N_le_O).symm, (Fin.castLEEmb hno).toEmbedding, a,
       fun i ↦ _⟩
   ext
   simp only [snf, Basis.map_apply, Submodule.comapSubtypeEquivOfLe_symm_apply,
-    Submodule.coe_smul_of_tower, RelEmbedding.coe_toEmbedding]
+    Submodule.coe_smul_of_tower, RelEmbedding.coe_toEmbedding, Fin.castLEEmb_apply]
 #align submodule.smith_normal_form_of_le Submodule.smithNormalFormOfLE
 
 /-- If `M` is finite free over a PID `R`, then any submodule `N` is free

@@ -220,7 +220,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
     refine' Finset.sum_congr rfl fun i _ ↦ _
     rw [← mul_smul, ← hc]
     rfl
-  -- We found an `y` and an `a`!
+  -- We found a `y` and an `a`!
   refine' ⟨y', y'M, a, a_smul_y'.symm ▸ yN, _⟩
   have ϕy'_eq : ϕ ⟨y', y'M⟩ = 1 :=
     mul_left_cancel₀ a_zero

Changes are of the form

• some_tactic at h⊢ -> some_tactic at h ⊢
• some_tactic at h -> some_tactic at h

@@ -237,7 +237,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
   have M'_le_M : M' ≤ M := M.map_subtype_le (LinearMap.ker ϕ)
   have N'_le_M' : N' ≤ M' := by
     intro x hx
-    simp only [mem_map, LinearMap.mem_ker] at hx⊢
+    simp only [mem_map, LinearMap.mem_ker] at hx ⊢
     obtain ⟨⟨x, xN⟩, hx, rfl⟩ := hx
     exact ⟨⟨x, N_le_M xN⟩, hx, rfl⟩
   have N'_le_N : N' ≤ N := N.map_subtype_le (LinearMap.ker (ϕ.comp (ofLe N_le_M)))

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -166,16 +166,16 @@ but must also feed in a basis for `M` using `basis_of_pid` to keep the induction
 theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Module R O]
     (M N : Submodule R O) (b'M : Basis ι R M) (N_bot : N ≠ ⊥) (N_le_M : N ≤ M) :
     ∃ y ∈ M,
-      ∃ (a : R)(_ : a • y ∈ N),
+      ∃ (a : R) (_ : a • y ∈ N),
         ∃ M' ≤ M,
           ∃ N' ≤ N,
-            ∃ (_N'_le_M' : N' ≤ M')(_y_ortho_M' :
-              ∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0)(_ay_ortho_N' :
+            ∃ (_N'_le_M' : N' ≤ M') (_y_ortho_M' :
+              ∀ (c : R) (z : O), z ∈ M' → c • y + z = 0 → c = 0) (_ay_ortho_N' :
               ∀ (c : R) (z : O), z ∈ N' → c • a • y + z = 0 → c = 0),
               ∀ (n') (bN' : Basis (Fin n') R N'),
                 ∃ bN : Basis (Fin (n' + 1)) R N,
                   ∀ (m') (hn'm' : n' ≤ m') (bM' : Basis (Fin m') R M'),
-                    ∃ (hnm : n' + 1 ≤ m' + 1)(bM : Basis (Fin (m' + 1)) R M),
+                    ∃ (hnm : n' + 1 ≤ m' + 1) (bM : Basis (Fin (m' + 1)) R M),
                       ∀ (as : Fin n' → R)
                         (_h : ∀ i : Fin n', (bN' i : O) = as i • (bM' (Fin.castLE hn'm' i) : O)),
                         ∃ as' : Fin (n' + 1) → R,
@@ -452,7 +452,7 @@ This is a strengthening of `Submodule.basisOfPidOfLE`.
 -/
 theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M) (N O : Submodule R M)
     (N_le_O : N ≤ O) :
-    ∃ (n o : ℕ)(hno : n ≤ o)(bO : Basis (Fin o) R O)(bN : Basis (Fin n) R N)(a : Fin n → R),
+    ∃ (n o : ℕ) (hno : n ≤ o) (bO : Basis (Fin o) R O) (bN : Basis (Fin n) R N) (a : Fin n → R),
       ∀ i, (bN i : M) = a i • bO (Fin.castLE hno i) := by
   cases nonempty_fintype ι
   revert N
@@ -553,7 +553,7 @@ The definitions `Ideal.ringBasis`, `Ideal.selfBasis`, `Ideal.smithCoeffs` are (n
 choices of values for this existential quantifier.
 -/
 theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
-    ∃ (b' : Basis ι R S)(a : ι → R)(ab' : Basis ι R I), ∀ i, (ab' i : S) = a i • b' i := by
+    ∃ (b' : Basis ι R S) (a : ι → R) (ab' : Basis ι R I), ∀ i, (ab' i : S) = a i • b' i := by
   cases nonempty_fintype ι
   let ⟨bS, bI, f, a, snf⟩ := I.smithNormalForm b hI
   let e : Fin (Fintype.card ι) ≃ ι :=

linear_algebra.free_module.pid@210657c4ea4a4a7b234392f70a3a2a83346dfa90..d87199d51218d36a0a42c66c82d147b5a7ff87b3

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Anne Baanen
 
 ! This file was ported from Lean 3 source module linear_algebra.free_module.pid
-! leanprover-community/mathlib commit 210657c4ea4a4a7b234392f70a3a2a83346dfa90
+! leanprover-community/mathlib commit d87199d51218d36a0a42c66c82d147b5a7ff87b3
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -623,6 +623,9 @@ theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   simp [hi]
 #align ideal.smith_coeffs_ne_zero Ideal.smithCoeffs_ne_zero
 
+-- porting note: can be inferred in Lean 4 so no longer necessary
+#noalign has_quotient.quotient.module
+
 end Ideal
 
 end SmithNormal

Now that leanprover/lean4#2210 has been merged, this PR:

• removes all the set_option synthInstance.etaExperiment true commands (and some etaExperiment% term elaborators)
• removes many but not quite all set_option maxHeartbeats commands
• makes various other changes required to cope with leanprover/lean4#2210.

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Matthew Ballard <matt@mrb.email>

@@ -64,7 +64,6 @@ variable {ι : Type _} (b : Basis ι R M)
 
 open Submodule.IsPrincipal Submodule
 
-set_option synthInstance.etaExperiment true in -- Porting note : added
 theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodule R M}
     {ϕ : M →ₗ[R] R} (hϕ : ∀ ψ : M →ₗ[R] R, ¬N.map ϕ < N.map ψ) [(N.map ϕ).IsPrincipal]
     (hgen : generator (N.map ϕ) = (0 : R)) : N = ⊥ := by
@@ -78,7 +77,6 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
       ⟨x, hx, rfl⟩
 #align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zero
 
-set_option synthInstance.etaExperiment true in
 theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
     (hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (hgen : generator (ϕ.submoduleImage N) = 0) : N = ⊥ := by
@@ -101,7 +99,6 @@ variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal Set Submodule
 
-set_option synthInstance.etaExperiment true in
 theorem dvd_generator_iff {I : Ideal R} [I.IsPrincipal] {x : R} (hx : x ∈ I) :
     x ∣ generator I ↔ I = Ideal.span {x} := by
   conv_rhs => rw [← span_singleton_generator I]
@@ -122,7 +119,6 @@ variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal
 
-set_option synthInstance.etaExperiment true in -- Porting note : added
 theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤ O) {ϕ : O →ₗ[R] R}
     (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (y : M) (yN : y ∈ N)
@@ -156,7 +152,6 @@ theorem generator_maximal_submoduleImage_dvd {N O : Submodule R M} (hNO : N ≤
   · exact subset_span (mem_insert _ _)
 #align generator_maximal_submodule_image_dvd generator_maximal_submoduleImage_dvd
 
-set_option synthInstance.etaExperiment true in
 /-- The induction hypothesis of `Submodule.basisOfPid` and `Submodule.smithNormalForm`.
 
 Basically, it says: let `N ≤ M` be a pair of submodules, then we can find a pair of
@@ -357,7 +352,6 @@ noncomputable def Submodule.basisOfPidOfLESpan {ι : Type _} [Finite ι] {b : ι
   Submodule.basisOfPidOfLE le (Basis.span hb)
 #align submodule.basis_of_pid_of_le_span Submodule.basisOfPidOfLESpan
 
-set_option synthInstance.etaExperiment true in -- Porting note: added
 /-- A finite type torsion free module over a PID admits a basis. -/
 noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M}
     (hs : span R (range s) = ⊤) [NoZeroSMulDivisors R M] : Σn : ℕ, Basis (Fin n) R M := by
@@ -525,7 +519,6 @@ section Ideal
 
 variable {S : Type _} [CommRing S] [IsDomain S] [Algebra R S]
 
-set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -548,7 +541,6 @@ noncomputable def Ideal.smithNormalForm [Fintype ι] (b : Basis ι R S) (I : Ide
 
 variable [Finite ι]
 
-set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -575,7 +567,6 @@ theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I
           Submodule.restrictScalarsEquiv_apply, Basis.coe_reindex, (· ∘ ·)]⟩
 #align ideal.exists_smith_normal_form Ideal.exists_smith_normal_form
 
-set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -588,7 +579,6 @@ noncomputable def Ideal.ringBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   (Ideal.exists_smith_normal_form b I hI).choose
 #align ideal.ring_basis Ideal.ringBasis
 
-set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -601,7 +591,6 @@ noncomputable def Ideal.selfBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose_spec.choose
 #align ideal.self_basis Ideal.selfBasis
 
-set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -614,7 +603,6 @@ noncomputable def Ideal.smithCoeffs (b : Basis ι R S) (I : Ideal S) (hI : I ≠
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose
 #align ideal.smith_coeffs Ideal.smithCoeffs
 
-set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -626,7 +614,6 @@ theorem Ideal.selfBasis_def (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose_spec.choose_spec
 #align ideal.self_basis_def Ideal.selfBasis_def
 
-set_option synthInstance.etaExperiment true in
 @[simp]
 theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) (i) :
     Ideal.smithCoeffs b I hI i ≠ 0 := by

@@ -105,7 +105,8 @@ set_option synthInstance.etaExperiment true in
 theorem dvd_generator_iff {I : Ideal R} [I.IsPrincipal] {x : R} (hx : x ∈ I) :
     x ∣ generator I ↔ I = Ideal.span {x} := by
   conv_rhs => rw [← span_singleton_generator I]
-  erw [Ideal.span_singleton_eq_span_singleton, ← dvd_dvd_iff_associated, ← mem_iff_generator_dvd]
+  rw [Ideal.submodule_span_eq, Ideal.span_singleton_eq_span_singleton, ← dvd_dvd_iff_associated,
+    ← mem_iff_generator_dvd]
   exact ⟨fun h ↦ ⟨hx, h⟩, fun h ↦ h.2⟩
 #align dvd_generator_iff dvd_generator_iff
 
@@ -542,7 +543,7 @@ noncomputable def Ideal.smithNormalForm [Fintype ι] (b : Basis ι R S) (I : Ide
   let e : Fin n ≃ Fin (Fintype.card ι) := Fintype.equivOfCardEq (by rw [eq, Fintype.card_fin])
   ⟨bS, bI.reindex e, e.symm.toEmbedding.trans f, a ∘ e.symm, fun i ↦ by
     simp only [snf, Basis.coe_reindex, Function.Embedding.trans_apply, Equiv.toEmbedding_apply,
-      (·∘·)]⟩
+      (· ∘ ·)]⟩
 #align ideal.smith_normal_form Ideal.smithNormalForm
 
 variable [Finite ι]
@@ -571,7 +572,7 @@ theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I
     ⟨bS, a ∘ e.symm, (bI.reindex e).map ((restrictScalarsEquiv R S _ _).restrictScalars R),
       fun i ↦ by
         simp only [snf, fe, Basis.map_apply, LinearEquiv.restrictScalars_apply R,
-          Submodule.restrictScalarsEquiv_apply, Basis.coe_reindex, (·∘·)]⟩
+          Submodule.restrictScalarsEquiv_apply, Basis.coe_reindex, (· ∘ ·)]⟩
 #align ideal.exists_smith_normal_form Ideal.exists_smith_normal_form
 
 set_option synthInstance.etaExperiment true in

@@ -271,8 +271,7 @@ theorem Submodule.basis_of_pid_aux [Finite ι] {O : Type _} [AddCommGroup O] [Mo
       refine' ⟨-b, Submodule.mem_map.mpr ⟨⟨_, N.sub_mem zN (N.smul_mem b yN)⟩, _, _⟩⟩
       · refine' LinearMap.mem_ker.mpr (show ϕ (⟨z, N_le_M zN⟩ - b • ⟨y, N_le_M yN⟩) = 0 from _)
         rw [LinearMap.map_sub, LinearMap.map_smul, hb, ϕy_eq, smul_eq_mul, mul_comm, sub_self]
-      · simp only [sub_eq_add_neg, neg_smul]
-        rfl
+      · simp only [sub_eq_add_neg, neg_smul, coeSubtype]
   -- And extend a basis for `M'` with `y'`
   intro m' hn'm' bM'
   refine' ⟨Nat.succ_le_succ hn'm', _, _⟩
@@ -307,7 +306,7 @@ theorem Submodule.nonempty_basis_of_pid {ι : Type _} [Finite ι] (b : Basis ι
   haveI := Classical.decEq M
   cases nonempty_fintype ι
   induction' N using inductionOnRank with N ih
-  exact b
+  · exact b
   let b' := (b.reindex (Fintype.equivFin ι)).map (LinearEquiv.ofTop _ rfl).symm
   by_cases N_bot : N = ⊥
   · subst N_bot
@@ -372,10 +371,8 @@ noncomputable def Module.basisOfFiniteTypeTorsionFree [Fintype ι] {s : ι → M
     let N := span R (range <| (s ∘ (fun x => x) : I → M))
     -- same as `span R (s '' I)` but more convenient
     let _sI : I → N := fun i ↦ ⟨s i.1, subset_span (mem_range_self i)⟩
-    -- `s` restricted to `I`
-    let sI_basis : Basis I R N
     -- `s` restricted to `I` is a basis of `N`
-    exact Basis.span indepI
+    let sI_basis : Basis I R N := Basis.span indepI
     -- Our first goal is to build `A ≠ 0` such that `A • M ⊆ N`
     have exists_a : ∀ i : ι, ∃ a : R, a ≠ 0 ∧ a • s i ∈ N := by
       intro i
@@ -465,7 +462,7 @@ theorem Submodule.exists_smith_normal_form_of_le [Finite ι] (b : Basis ι R M)
   cases nonempty_fintype ι
   revert N
   induction' O using inductionOnRank with M0 ih
-  exact b
+  · exact b
   intro N N_le_M0
   obtain ⟨m, b'M⟩ := M0.basisOfPid b
   by_cases N_bot : N = ⊥
@@ -564,17 +561,17 @@ choices of values for this existential quantifier.
 -/
 theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
     ∃ (b' : Basis ι R S)(a : ι → R)(ab' : Basis ι R I), ∀ i, (ab' i : S) = a i • b' i := by
-  cases nonempty_fintype ι;
-    exact
-      let ⟨bS, bI, f, a, snf⟩ := I.smithNormalForm b hI
-      let e : Fin (Fintype.card ι) ≃ ι :=
-        Equiv.ofBijective f
-          ((Fintype.bijective_iff_injective_and_card f).mpr ⟨f.injective, Fintype.card_fin _⟩)
-      have fe : ∀ i, f (e.symm i) = i := e.apply_symm_apply
-      ⟨bS, a ∘ e.symm, (bI.reindex e).map ((restrictScalarsEquiv R S _ _).restrictScalars R),
-        fun i ↦ by
-          simp only [snf, fe, Basis.map_apply, LinearEquiv.restrictScalars_apply R,
-            Submodule.restrictScalarsEquiv_apply, Basis.coe_reindex, (·∘·)]⟩
+  cases nonempty_fintype ι
+  let ⟨bS, bI, f, a, snf⟩ := I.smithNormalForm b hI
+  let e : Fin (Fintype.card ι) ≃ ι :=
+    Equiv.ofBijective f
+      ((Fintype.bijective_iff_injective_and_card f).mpr ⟨f.injective, Fintype.card_fin _⟩)
+  have fe : ∀ i, f (e.symm i) = i := e.apply_symm_apply
+  exact
+    ⟨bS, a ∘ e.symm, (bI.reindex e).map ((restrictScalarsEquiv R S _ _).restrictScalars R),
+      fun i ↦ by
+        simp only [snf, fe, Basis.map_apply, LinearEquiv.restrictScalars_apply R,
+          Submodule.restrictScalarsEquiv_apply, Basis.coe_reindex, (·∘·)]⟩
 #align ideal.exists_smith_normal_form Ideal.exists_smith_normal_form
 
 set_option synthInstance.etaExperiment true in

This is to fix timeouts in https://github.com/leanprover-community/mathlib4/pull/3552.

See discussion at https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/!4.233552.20.28LinearAlgebra.2EMatrix.2EToLin.29.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -78,6 +78,7 @@ theorem eq_bot_of_generator_maximal_map_eq_zero (b : Basis ι R M) {N : Submodul
       ⟨x, hx, rfl⟩
 #align eq_bot_of_generator_maximal_map_eq_zero eq_bot_of_generator_maximal_map_eq_zero
 
+set_option synthInstance.etaExperiment true in
 theorem eq_bot_of_generator_maximal_submoduleImage_eq_zero {N O : Submodule R M} (b : Basis ι R O)
     (hNO : N ≤ O) {ϕ : O →ₗ[R] R} (hϕ : ∀ ψ : O →ₗ[R] R, ¬ϕ.submoduleImage N < ψ.submoduleImage N)
     [(ϕ.submoduleImage N).IsPrincipal] (hgen : generator (ϕ.submoduleImage N) = 0) : N = ⊥ := by
@@ -100,6 +101,7 @@ variable {M : Type _} [AddCommGroup M] [Module R M] {b : ι → M}
 
 open Submodule.IsPrincipal Set Submodule
 
+set_option synthInstance.etaExperiment true in
 theorem dvd_generator_iff {I : Ideal R} [I.IsPrincipal] {x : R} (hx : x ∈ I) :
     x ∣ generator I ↔ I = Ideal.span {x} := by
   conv_rhs => rw [← span_singleton_generator I]
@@ -525,6 +527,7 @@ section Ideal
 
 variable {S : Type _} [CommRing S] [IsDomain S] [Algebra R S]
 
+set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -547,6 +550,7 @@ noncomputable def Ideal.smithNormalForm [Fintype ι] (b : Basis ι R S) (I : Ide
 
 variable [Finite ι]
 
+set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -573,6 +577,7 @@ theorem Ideal.exists_smith_normal_form (b : Basis ι R S) (I : Ideal S) (hI : I
             Submodule.restrictScalarsEquiv_apply, Basis.coe_reindex, (·∘·)]⟩
 #align ideal.exists_smith_normal_form Ideal.exists_smith_normal_form
 
+set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -585,6 +590,7 @@ noncomputable def Ideal.ringBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   (Ideal.exists_smith_normal_form b I hI).choose
 #align ideal.ring_basis Ideal.ringBasis
 
+set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -597,6 +603,7 @@ noncomputable def Ideal.selfBasis (b : Basis ι R S) (I : Ideal S) (hI : I ≠ 
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose_spec.choose
 #align ideal.self_basis Ideal.selfBasis
 
+set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -609,6 +616,7 @@ noncomputable def Ideal.smithCoeffs (b : Basis ι R S) (I : Ideal S) (hI : I ≠
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose
 #align ideal.smith_coeffs Ideal.smithCoeffs
 
+set_option synthInstance.etaExperiment true in
 /-- If `S` a finite-dimensional ring extension of a PID `R` which is free as an `R`-module,
 then any nonzero `S`-ideal `I` is free as an `R`-submodule of `S`, and we can
 find a basis for `S` and `I` such that the inclusion map is a square diagonal
@@ -620,6 +628,7 @@ theorem Ideal.selfBasis_def (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) :
   (Ideal.exists_smith_normal_form b I hI).choose_spec.choose_spec.choose_spec
 #align ideal.self_basis_def Ideal.selfBasis_def
 
+set_option synthInstance.etaExperiment true in
 @[simp]
 theorem Ideal.smithCoeffs_ne_zero (b : Basis ι R S) (I : Ideal S) (hI : I ≠ ⊥) (i) :
     Ideal.smithCoeffs b I hI i ≠ 0 := by

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>