algebra.module.torsion ⟷ Mathlib.Algebra.Module.Torsion

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

@@ -936,7 +936,7 @@ theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
   · obtain ⟨n, h0, hn⟩ := (isOfFinAddOrder_iff_nsmul_eq_zero x).mp (h x)
     exact
       ⟨⟨n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt <| int.coe_nat_pos.mpr h0⟩,
-        (coe_nat_zsmul _ _).trans hn⟩
+        (natCast_zsmul _ _).trans hn⟩
   · rw [isOfFinAddOrder_iff_nsmul_eq_zero]
     obtain ⟨n, hn⟩ := @h x
     exact exists_nsmul_eq_zero_of_zsmul_eq_zero (nonZeroDivisors.coe_ne_zero n) hn

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

@@ -112,9 +112,9 @@ theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDiv
     torsionOf R M m = ⊥ ↔ m ≠ 0 :=
   by
   refine' ⟨fun h contra => _, fun h => (Submodule.eq_bot_iff _).mpr fun r hr => _⟩
-  · rw [contra, torsion_of_zero] at h 
+  · rw [contra, torsion_of_zero] at h
     exact bot_ne_top.symm h
-  · rw [mem_torsion_of_iff, smul_eq_zero] at hr 
+  · rw [mem_torsion_of_iff, smul_eq_zero] at hr
     tauto
 #align ideal.torsion_of_eq_bot_iff_of_no_zero_smul_divisors Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisors
 -/
@@ -128,7 +128,7 @@ theorem CompleteLattice.Independent.linear_independent' {ι R M : Type _} {v : 
   by
   refine' linear_independent_iff_not_smul_mem_span.mpr fun i r hi => _
   replace hv := complete_lattice.independent_def.mp hv i
-  simp only [iSup_subtype', ← Submodule.span_range_eq_iSup, disjoint_iff] at hv 
+  simp only [iSup_subtype', ← Submodule.span_range_eq_iSup, disjoint_iff] at hv
   have : r • v i ∈ ⊥ := by
     rw [← hv, Submodule.mem_inf]
     refine' ⟨submodule.mem_span_singleton.mpr ⟨r, rfl⟩, _⟩
@@ -342,7 +342,7 @@ theorem torsionBy_le_torsionBy_of_dvd (a b : R) (dvd : a ∣ b) : torsionBy R M
 #print Submodule.torsionBy_one /-
 @[simp]
 theorem torsionBy_one : torsionBy R M 1 = ⊥ :=
-  eq_bot_iff.mpr fun _ h => by rw [mem_torsion_by_iff, one_smul] at h ; exact h
+  eq_bot_iff.mpr fun _ h => by rw [mem_torsion_by_iff, one_smul] at h; exact h
 #align submodule.torsion_by_one Submodule.torsionBy_one
 -/
 
@@ -476,7 +476,7 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
       rw [mem_infi]; intro j; rw [mem_infi]; intro hj
       by_cases ij : j = i
       · rw [ij]; exact Ideal.mul_mem_right _ _ ha
-      · have := coe_mem (μ i); simp only [mem_infi] at this 
+      · have := coe_mem (μ i); simp only [mem_infi] at this
         exact Ideal.mul_mem_left _ _ (this j hj ij)
     · simp_rw [coe_mk]; rw [← Finset.sum_smul, hμ, one_smul]
 #align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf
@@ -559,7 +559,7 @@ theorem torsionBy_isInternal {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCo
     DirectSum.IsInternal fun i : S => torsionBy R M <| q i :=
   by
   rw [← Module.isTorsionBySet_span_singleton_iff, Ideal.submodule_span_eq, ←
-    Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf] at hM 
+    Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf] at hM
   convert
     torsion_by_set_is_internal
       (fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij) hM
@@ -581,7 +581,7 @@ def IsTorsionBySet.hasSMul : SMul (R ⧸ I) M
       by
       show b₁ • x = b₂ • x
       have : (-b₁ + b₂) • x = 0 := @hM x ⟨_, quotient_add_group.left_rel_apply.mp h⟩
-      rw [add_smul, neg_smul, neg_add_eq_zero] at this 
+      rw [add_smul, neg_smul, neg_add_eq_zero] at this
       exact this
 #align module.is_torsion_by_set.has_smul Module.IsTorsionBySet.hasSMul
 -/
@@ -783,7 +783,7 @@ theorem noZeroSMulDivisors_iff_torsion_eq_bot : NoZeroSMulDivisors R M ↔ torsi
   constructor <;> intro h
   · haveI : NoZeroSMulDivisors R M := h
     rw [eq_bot_iff]; rintro x ⟨a, hax⟩
-    change (a : R) • x = 0 at hax 
+    change (a : R) • x = 0 at hax
     cases' eq_zero_or_eq_zero_of_smul_eq_zero hax with h0 h0
     · exfalso; exact nonZeroDivisors.coe_ne_zero a h0; · exact h0
   ·
@@ -811,7 +811,7 @@ theorem torsion_eq_bot : torsion R (M ⧸ torsion R M) = ⊥ :=
   eq_bot_iff.mpr fun z =>
     Quotient.inductionOn' z fun x ⟨a, hax⟩ =>
       by
-      rw [Quotient.mk''_eq_mk', ← quotient.mk_smul, quotient.mk_eq_zero] at hax 
+      rw [Quotient.mk''_eq_mk', ← quotient.mk_smul, quotient.mk_eq_zero] at hax
       rw [mem_bot, Quotient.mk''_eq_mk', quotient.mk_eq_zero]
       cases' hax with b h
       exact ⟨b * a, (mul_smul _ _ _).trans h⟩
@@ -900,10 +900,10 @@ theorem torsionBy_eq_span_singleton {R : Type _} [CommRing R] (a b : R) (ha : a
   by
   ext x; rw [mem_torsion_by_iff, mem_span_singleton]
   obtain ⟨x, rfl⟩ := mk_surjective x; constructor <;> intro h
-  · rw [← mk_eq_mk, ← quotient.mk_smul, quotient.mk_eq_zero, mem_span_singleton] at h 
+  · rw [← mk_eq_mk, ← quotient.mk_smul, quotient.mk_eq_zero, mem_span_singleton] at h
     obtain ⟨c, h⟩ := h;
     rw [smul_eq_mul, smul_eq_mul, mul_comm, mul_assoc, mul_cancel_left_mem_nonZeroDivisors ha,
-      mul_comm] at h 
+      mul_comm] at h
     use c; rw [← h, ← mk_eq_mk, ← quotient.mk_smul, smul_eq_mul, mk_eq_mk]
   · obtain ⟨c, h⟩ := h
     rw [← h, smul_comm, ← mk_eq_mk, ← quotient.mk_smul,

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

@@ -3,12 +3,12 @@ Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Pierre-Alexandre Bazin
 -/
-import Mathbin.Algebra.DirectSum.Module
-import Mathbin.Algebra.Module.BigOperators
-import Mathbin.LinearAlgebra.Isomorphisms
-import Mathbin.GroupTheory.Torsion
-import Mathbin.RingTheory.Coprime.Ideal
-import Mathbin.RingTheory.Finiteness
+import Algebra.DirectSum.Module
+import Algebra.Module.BigOperators
+import LinearAlgebra.Isomorphisms
+import GroupTheory.Torsion
+import RingTheory.Coprime.Ideal
+import RingTheory.Finiteness
 
 #align_import algebra.module.torsion from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
 

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

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Pierre-Alexandre Bazin
-
-! This file was ported from Lean 3 source module algebra.module.torsion
-! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.DirectSum.Module
 import Mathbin.Algebra.Module.BigOperators
@@ -15,6 +10,8 @@ import Mathbin.GroupTheory.Torsion
 import Mathbin.RingTheory.Coprime.Ideal
 import Mathbin.RingTheory.Finiteness
 
+#align_import algebra.module.torsion from "leanprover-community/mathlib"@"0b7c740e25651db0ba63648fbae9f9d6f941e31b"
+
 /-!
 # Torsion submodules
 

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

@@ -82,19 +82,24 @@ def torsionOf (x : M) : Ideal R :=
 #align ideal.torsion_of Ideal.torsionOf
 -/
 
+#print Ideal.torsionOf_zero /-
 @[simp]
 theorem torsionOf_zero : torsionOf R M (0 : M) = ⊤ := by simp [torsion_of]
 #align ideal.torsion_of_zero Ideal.torsionOf_zero
+-/
 
 variable {R M}
 
+#print Ideal.mem_torsionOf_iff /-
 @[simp]
 theorem mem_torsionOf_iff (x : M) (a : R) : a ∈ torsionOf R M x ↔ a • x = 0 :=
   Iff.rfl
 #align ideal.mem_torsion_of_iff Ideal.mem_torsionOf_iff
+-/
 
 variable (R)
 
+#print Ideal.torsionOf_eq_top_iff /-
 @[simp]
 theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 :=
   by
@@ -102,7 +107,9 @@ theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 :=
   rw [← one_smul R m, ← mem_torsion_of_iff m (1 : R), h]
   exact Submodule.mem_top
 #align ideal.torsion_of_eq_top_iff Ideal.torsionOf_eq_top_iff
+-/
 
+#print Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisors /-
 @[simp]
 theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDivisors R M] (m : M) :
     torsionOf R M m = ⊥ ↔ m ≠ 0 :=
@@ -113,7 +120,9 @@ theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDiv
   · rw [mem_torsion_of_iff, smul_eq_zero] at hr 
     tauto
 #align ideal.torsion_of_eq_bot_iff_of_no_zero_smul_divisors Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisors
+-/
 
+#print Ideal.CompleteLattice.Independent.linear_independent' /-
 /-- See also `complete_lattice.independent.linear_independent` which provides the same conclusion
 but requires the stronger hypothesis `no_zero_smul_divisors R M`. -/
 theorem CompleteLattice.Independent.linear_independent' {ι R M : Type _} {v : ι → M} [Ring R]
@@ -132,6 +141,7 @@ theorem CompleteLattice.Independent.linear_independent' {ι R M : Type _} {v : 
   rw [← Submodule.mem_bot R, ← h_ne_zero i]
   simpa using this
 #align ideal.complete_lattice.independent.linear_independent' Ideal.CompleteLattice.Independent.linear_independent'
+-/
 
 end TorsionOf
 
@@ -149,12 +159,14 @@ noncomputable def quotTorsionOfEquivSpanSingleton (x : M) : (R ⧸ torsionOf R M
 
 variable {R M}
 
+#print Ideal.quotTorsionOfEquivSpanSingleton_apply_mk /-
 @[simp]
 theorem quotTorsionOfEquivSpanSingleton_apply_mk (x : M) (a : R) :
     quotTorsionOfEquivSpanSingleton R M x (Submodule.Quotient.mk a) =
       a • ⟨x, Submodule.mem_span_singleton_self x⟩ :=
   rfl
 #align ideal.quot_torsion_of_equiv_span_singleton_apply_mk Ideal.quotTorsionOfEquivSpanSingleton_apply_mk
+-/
 
 end
 
@@ -185,6 +197,7 @@ def torsionBySet (s : Set R) : Submodule R M :=
 #align submodule.torsion_by_set Submodule.torsionBySet
 -/
 
+#print Submodule.torsion' /-
 /-- The `S`-torsion submodule, containing all elements `x` of `M` such that `a • x = 0` for some
 `a` in `S`. -/
 @[simps]
@@ -196,6 +209,7 @@ def torsion' (S : Type _) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S
     ⟨b * a, by rw [smul_add, mul_smul, mul_comm, mul_smul, hx, hy, smul_zero, smul_zero, add_zero]⟩
   smul_mem' := fun a x ⟨b, h⟩ => ⟨b, by rw [smul_comm, h, smul_zero]⟩
 #align submodule.torsion' Submodule.torsion'
+-/
 
 #print Submodule.torsion /-
 /-- The torsion submodule, containing all elements `x` of `M` such that  `a • x = 0` for some
@@ -262,22 +276,28 @@ theorem smul_torsionBy (x : torsionBy R M a) : a • x = 0 :=
 #align submodule.smul_torsion_by Submodule.smul_torsionBy
 -/
 
+#print Submodule.smul_coe_torsionBy /-
 @[simp]
 theorem smul_coe_torsionBy (x : torsionBy R M a) : a • (x : M) = 0 :=
   x.Prop
 #align submodule.smul_coe_torsion_by Submodule.smul_coe_torsionBy
+-/
 
+#print Submodule.mem_torsionBy_iff /-
 @[simp]
 theorem mem_torsionBy_iff (x : M) : x ∈ torsionBy R M a ↔ a • x = 0 :=
   Iff.rfl
 #align submodule.mem_torsion_by_iff Submodule.mem_torsionBy_iff
+-/
 
+#print Submodule.mem_torsionBySet_iff /-
 @[simp]
 theorem mem_torsionBySet_iff (x : M) : x ∈ torsionBySet R M s ↔ ∀ a : s, (a : R) • x = 0 :=
   by
   refine' ⟨fun h ⟨a, ha⟩ => mem_Inf.mp h _ (Set.mem_image_of_mem _ ha), fun h => mem_Inf.mpr _⟩
   rintro _ ⟨a, ha, rfl⟩; exact h ⟨a, ha⟩
 #align submodule.mem_torsion_by_set_iff Submodule.mem_torsionBySet_iff
+-/
 
 #print Submodule.torsionBySet_singleton_eq /-
 @[simp]
@@ -289,10 +309,12 @@ theorem torsionBySet_singleton_eq : torsionBySet R M {a} = torsionBy R M a :=
 #align submodule.torsion_by_singleton_eq Submodule.torsionBySet_singleton_eq
 -/
 
+#print Submodule.torsionBySet_le_torsionBySet_of_subset /-
 theorem torsionBySet_le_torsionBySet_of_subset {s t : Set R} (st : s ⊆ t) :
     torsionBySet R M t ≤ torsionBySet R M s :=
   sInf_le_sInf fun _ ⟨a, ha, h⟩ => ⟨a, st ha, h⟩
 #align submodule.torsion_by_set_le_torsion_by_set_of_subset Submodule.torsionBySet_le_torsionBySet_of_subset
+-/
 
 #print Submodule.torsionBySet_eq_torsionBySet_span /-
 /-- Torsion by a set is torsion by the ideal generated by it. -/
@@ -311,24 +333,30 @@ theorem torsionBySet_span_singleton_eq : torsionBySet R M (R ∙ a) = torsionBy
 #align submodule.torsion_by_span_singleton_eq Submodule.torsionBySet_span_singleton_eq
 -/
 
+#print Submodule.torsionBy_le_torsionBy_of_dvd /-
 theorem torsionBy_le_torsionBy_of_dvd (a b : R) (dvd : a ∣ b) : torsionBy R M a ≤ torsionBy R M b :=
   by
   rw [← torsion_by_span_singleton_eq, ← torsion_by_singleton_eq]
   apply torsion_by_set_le_torsion_by_set_of_subset
   rintro c (rfl : c = b); exact ideal.mem_span_singleton.mpr dvd
 #align submodule.torsion_by_le_torsion_by_of_dvd Submodule.torsionBy_le_torsionBy_of_dvd
+-/
 
+#print Submodule.torsionBy_one /-
 @[simp]
 theorem torsionBy_one : torsionBy R M 1 = ⊥ :=
   eq_bot_iff.mpr fun _ h => by rw [mem_torsion_by_iff, one_smul] at h ; exact h
 #align submodule.torsion_by_one Submodule.torsionBy_one
+-/
 
+#print Submodule.torsionBySet_univ /-
 @[simp]
 theorem torsionBySet_univ : torsionBySet R M Set.univ = ⊥ :=
   by
   rw [eq_bot_iff, ← torsion_by_one, ← torsion_by_singleton_eq]
   exact torsion_by_set_le_torsion_by_set_of_subset fun _ _ => trivial
 #align submodule.torsion_by_univ Submodule.torsionBySet_univ
+-/
 
 end Submodule
 
@@ -336,34 +364,44 @@ open Submodule
 
 namespace Module
 
+#print Module.isTorsionBySet_singleton_iff /-
 @[simp]
 theorem isTorsionBySet_singleton_iff : IsTorsionBySet R M {a} ↔ IsTorsionBy R M a :=
   by
   refine' ⟨fun h x => @h _ ⟨_, Set.mem_singleton _⟩, fun h x => _⟩
   rintro ⟨b, rfl : b = a⟩; exact @h _
 #align module.is_torsion_by_singleton_iff Module.isTorsionBySet_singleton_iff
+-/
 
+#print Module.isTorsionBySet_iff_torsionBySet_eq_top /-
 theorem isTorsionBySet_iff_torsionBySet_eq_top :
     IsTorsionBySet R M s ↔ Submodule.torsionBySet R M s = ⊤ :=
   ⟨fun h => eq_top_iff.mpr fun _ _ => (mem_torsionBySet_iff _ _).mpr <| @h _, fun h x => by
     rw [← mem_torsion_by_set_iff, h]; trivial⟩
 #align module.is_torsion_by_set_iff_torsion_by_set_eq_top Module.isTorsionBySet_iff_torsionBySet_eq_top
+-/
 
+#print Module.isTorsionBy_iff_torsionBy_eq_top /-
 /-- A `a`-torsion module is a module whose `a`-torsion submodule is the full space. -/
 theorem isTorsionBy_iff_torsionBy_eq_top : IsTorsionBy R M a ↔ torsionBy R M a = ⊤ := by
   rw [← torsion_by_singleton_eq, ← is_torsion_by_singleton_iff,
     is_torsion_by_set_iff_torsion_by_set_eq_top]
 #align module.is_torsion_by_iff_torsion_by_eq_top Module.isTorsionBy_iff_torsionBy_eq_top
+-/
 
+#print Module.isTorsionBySet_iff_is_torsion_by_span /-
 theorem isTorsionBySet_iff_is_torsion_by_span :
     IsTorsionBySet R M s ↔ IsTorsionBySet R M (Ideal.span s) := by
   rw [is_torsion_by_set_iff_torsion_by_set_eq_top, is_torsion_by_set_iff_torsion_by_set_eq_top,
     torsion_by_set_eq_torsion_by_span]
 #align module.is_torsion_by_set_iff_is_torsion_by_span Module.isTorsionBySet_iff_is_torsion_by_span
+-/
 
+#print Module.isTorsionBySet_span_singleton_iff /-
 theorem isTorsionBySet_span_singleton_iff : IsTorsionBySet R M (R ∙ a) ↔ IsTorsionBy R M a :=
   (isTorsionBySet_iff_is_torsion_by_span _).symm.trans <| isTorsionBySet_singleton_iff _
 #align module.is_torsion_by_span_singleton_iff Module.isTorsionBySet_span_singleton_iff
+-/
 
 end Module
 
@@ -371,23 +409,31 @@ namespace Submodule
 
 open Module
 
+#print Submodule.torsionBySet_isTorsionBySet /-
 theorem torsionBySet_isTorsionBySet : IsTorsionBySet R (torsionBySet R M s) s := fun ⟨x, hx⟩ a =>
   Subtype.ext <| (mem_torsionBySet_iff _ _).mp hx a
 #align submodule.torsion_by_set_is_torsion_by_set Submodule.torsionBySet_isTorsionBySet
+-/
 
+#print Submodule.torsionBy_isTorsionBy /-
 /-- The `a`-torsion submodule is a `a`-torsion module. -/
 theorem torsionBy_isTorsionBy : IsTorsionBy R (torsionBy R M a) a := fun _ => smul_torsionBy _ _
 #align submodule.torsion_by_is_torsion_by Submodule.torsionBy_isTorsionBy
+-/
 
+#print Submodule.torsionBy_torsionBy_eq_top /-
 @[simp]
 theorem torsionBy_torsionBy_eq_top : torsionBy R (torsionBy R M a) a = ⊤ :=
   (isTorsionBy_iff_torsionBy_eq_top a).mp <| torsionBy_isTorsionBy a
 #align submodule.torsion_by_torsion_by_eq_top Submodule.torsionBy_torsionBy_eq_top
+-/
 
+#print Submodule.torsionBySet_torsionBySet_eq_top /-
 @[simp]
 theorem torsionBySet_torsionBySet_eq_top : torsionBySet R (torsionBySet R M s) s = ⊤ :=
   (isTorsionBySet_iff_torsionBySet_eq_top s).mp <| torsionBySet_isTorsionBySet s
 #align submodule.torsion_by_set_torsion_by_set_eq_top Submodule.torsionBySet_torsionBySet_eq_top
+-/
 
 variable (R M)
 
@@ -411,8 +457,7 @@ variable {ι : Type _} {p : ι → Ideal R} {S : Finset ι}
 
 variable (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
 
-include hp
-
+#print Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf /-
 theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
     (⨆ i ∈ S, torsionBySet R M <| p i) = torsionBySet R M ↑(⨅ i ∈ S, p i) :=
   by
@@ -438,7 +483,9 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
         exact Ideal.mul_mem_left _ _ (this j hj ij)
     · simp_rw [coe_mk]; rw [← Finset.sum_smul, hμ, one_smul]
 #align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf
+-/
 
+#print Submodule.supIndep_torsionBySet_ideal /-
 theorem supIndep_torsionBySet_ideal : S.SupIndep fun i => torsionBySet R M <| p i :=
   fun T hT i hi hiT =>
   by
@@ -450,13 +497,11 @@ theorem supIndep_torsionBySet_ideal : S.SupIndep fun i => torsionBySet R M <| p
   rw [← this, Ideal.sup_iInf_eq_top, top_coe, torsion_by_univ]
   intro j hj; apply hp hi (hT hj); rintro rfl; exact hiT hj
 #align submodule.sup_indep_torsion_by_ideal Submodule.supIndep_torsionBySet_ideal
-
-omit hp
+-/
 
 variable {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCoprime on q))
 
-include hq
-
+#print Submodule.iSup_torsionBy_eq_torsionBy_prod /-
 theorem iSup_torsionBy_eq_torsionBy_prod :
     (⨆ i ∈ S, torsionBy R M <| q i) = torsionBy R M (∏ i in S, q i) :=
   by
@@ -466,7 +511,9 @@ theorem iSup_torsionBy_eq_torsionBy_prod :
   · congr; ext : 1; congr; ext : 1; exact (torsion_by_span_singleton_eq _).symm
   · exact fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime _ _).mpr (hq hi hj ij)
 #align submodule.supr_torsion_by_eq_torsion_by_prod Submodule.iSup_torsionBy_eq_torsionBy_prod
+-/
 
+#print Submodule.supIndep_torsionBy /-
 theorem supIndep_torsionBy : S.SupIndep fun i => torsionBy R M <| q i :=
   by
   convert
@@ -474,6 +521,7 @@ theorem supIndep_torsionBy : S.SupIndep fun i => torsionBy R M <| q i :=
       (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij
   ext : 1; exact (torsion_by_span_singleton_eq _).symm
 #align submodule.sup_indep_torsion_by Submodule.supIndep_torsionBy
+-/
 
 end Coprime
 
@@ -491,6 +539,7 @@ open scoped BigOperators
 
 variable {ι : Type _} [DecidableEq ι] {S : Finset ι}
 
+#print Submodule.torsionBySet_isInternal /-
 /-- If the `p i` are pairwise coprime, a `⨅ i, p i`-torsion module is the internal direct sum of
 its `p i`-torsion submodules.-/
 theorem torsionBySet_isInternal {p : ι → Ideal R}
@@ -503,7 +552,9 @@ theorem torsionBySet_isInternal {p : ι → Ideal R}
       (iSup_torsionBySet_ideal_eq_torsionBySet_iInf hp).trans <|
         (Module.isTorsionBySet_iff_torsionBySet_eq_top _).mp hM)
 #align submodule.torsion_by_set_is_internal Submodule.torsionBySet_isInternal
+-/
 
+#print Submodule.torsionBy_isInternal /-
 /-- If the `q i` are pairwise coprime, a `∏ i, q i`-torsion module is the internal direct sum of
 its `q i`-torsion submodules.-/
 theorem torsionBy_isInternal {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCoprime on q))
@@ -517,6 +568,7 @@ theorem torsionBy_isInternal {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCo
       (fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij) hM
   ext : 1; exact (torsion_by_span_singleton_eq _).symm
 #align submodule.torsion_by_is_internal Submodule.torsionBy_isInternal
+-/
 
 end Submodule
 
@@ -524,8 +576,6 @@ namespace Module
 
 variable {I : Ideal R} (hM : IsTorsionBySet R M I)
 
-include hM
-
 #print Module.IsTorsionBySet.hasSMul /-
 /-- can't be an instance because hM can't be inferred -/
 def IsTorsionBySet.hasSMul : SMul (R ⧸ I) M
@@ -539,12 +589,14 @@ def IsTorsionBySet.hasSMul : SMul (R ⧸ I) M
 #align module.is_torsion_by_set.has_smul Module.IsTorsionBySet.hasSMul
 -/
 
+#print Module.IsTorsionBySet.mk_smul /-
 @[simp]
 theorem IsTorsionBySet.mk_smul (b : R) (x : M) :
     haveI := hM.has_smul
     Ideal.Quotient.mk I b • x = b • x :=
   rfl
 #align module.is_torsion_by_set.mk_smul Module.IsTorsionBySet.mk_smul
+-/
 
 #print Module.IsTorsionBySet.module /-
 /-- A `(R ⧸ I)`-module is a `R`-module which `is_torsion_by_set R M I`. -/
@@ -554,12 +606,12 @@ def IsTorsionBySet.module : Module (R ⧸ I) M :=
 #align module.is_torsion_by_set.module Module.IsTorsionBySet.module
 -/
 
+#print Module.IsTorsionBySet.isScalarTower /-
 instance IsTorsionBySet.isScalarTower {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M]
     [IsScalarTower S R R] : @IsScalarTower S (R ⧸ I) M _ (IsTorsionBySet.module hM).toSMul _
     where smul_assoc b d x := Quotient.inductionOn' d fun c => (smul_assoc b c x : _)
 #align module.is_torsion_by_set.is_scalar_tower Module.IsTorsionBySet.isScalarTower
-
-omit hM
+-/
 
 instance : Module (R ⧸ I) (M ⧸ I • (⊤ : Submodule R M)) :=
   IsTorsionBySet.module fun x r =>
@@ -575,11 +627,13 @@ namespace Submodule
 instance (I : Ideal R) : Module (R ⧸ I) (torsionBySet R M I) :=
   Module.IsTorsionBySet.module <| torsionBySet_isTorsionBySet I
 
+#print Submodule.torsionBySet.mk_smul /-
 @[simp]
 theorem torsionBySet.mk_smul (I : Ideal R) (b : R) (x : torsionBySet R M I) :
     Ideal.Quotient.mk I b • x = b • x :=
   rfl
 #align submodule.torsion_by_set.mk_smul Submodule.torsionBySet.mk_smul
+-/
 
 instance (I : Ideal R) {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M]
     [IsScalarTower S R R] : IsScalarTower S (R ⧸ I) (torsionBySet R M I) :=
@@ -590,11 +644,13 @@ instance (a : R) : Module (R ⧸ R ∙ a) (torsionBy R M a) :=
   Module.IsTorsionBySet.module <|
     (Module.isTorsionBySet_span_singleton_iff a).mpr <| torsionBy_isTorsionBy a
 
+#print Submodule.torsionBy.mk_smul /-
 @[simp]
 theorem torsionBy.mk_smul (a b : R) (x : torsionBy R M a) :
     Ideal.Quotient.mk (R ∙ a) b • x = b • x :=
   rfl
 #align submodule.torsion_by.mk_smul Submodule.torsionBy.mk_smul
+-/
 
 instance (a : R) {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M] [IsScalarTower S R R] :
     IsScalarTower S (R ⧸ R ∙ a) (torsionBy R M a) :=
@@ -614,15 +670,19 @@ variable [CommSemiring R] [AddCommMonoid M] [Module R M]
 
 variable (S : Type _) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S R M]
 
+#print Submodule.mem_torsion'_iff /-
 @[simp]
 theorem mem_torsion'_iff (x : M) : x ∈ torsion' R M S ↔ ∃ a : S, a • x = 0 :=
   Iff.rfl
 #align submodule.mem_torsion'_iff Submodule.mem_torsion'_iff
+-/
 
+#print Submodule.mem_torsion_iff /-
 @[simp]
 theorem mem_torsion_iff (x : M) : x ∈ torsion R M ↔ ∃ a : R⁰, a • x = 0 :=
   Iff.rfl
 #align submodule.mem_torsion_iff Submodule.mem_torsion_iff
+-/
 
 @[simps]
 instance : SMul S (torsion' R M S) :=
@@ -635,31 +695,41 @@ instance : DistribMulAction S (torsion' R M S) :=
 instance : SMulCommClass S R (torsion' R M S) :=
   ⟨fun s a x => Subtype.ext <| smul_comm _ _ _⟩
 
+#print Submodule.isTorsion'_iff_torsion'_eq_top /-
 /-- A `S`-torsion module is a module whose `S`-torsion submodule is the full space. -/
 theorem isTorsion'_iff_torsion'_eq_top : IsTorsion' M S ↔ torsion' R M S = ⊤ :=
   ⟨fun h => eq_top_iff.mpr fun _ _ => @h _, fun h x => by rw [← @mem_torsion'_iff R, h]; trivial⟩
 #align submodule.is_torsion'_iff_torsion'_eq_top Submodule.isTorsion'_iff_torsion'_eq_top
+-/
 
+#print Submodule.torsion'_isTorsion' /-
 /-- The `S`-torsion submodule is a `S`-torsion module. -/
 theorem torsion'_isTorsion' : IsTorsion' (torsion' R M S) S := fun ⟨x, ⟨a, h⟩⟩ => ⟨a, Subtype.ext h⟩
 #align submodule.torsion'_is_torsion' Submodule.torsion'_isTorsion'
+-/
 
+#print Submodule.torsion'_torsion'_eq_top /-
 @[simp]
 theorem torsion'_torsion'_eq_top : torsion' R (torsion' R M S) S = ⊤ :=
   (isTorsion'_iff_torsion'_eq_top S).mp <| torsion'_isTorsion' S
 #align submodule.torsion'_torsion'_eq_top Submodule.torsion'_torsion'_eq_top
+-/
 
+#print Submodule.torsion_torsion_eq_top /-
 /-- The torsion submodule of the torsion submodule (viewed as a module) is the full
 torsion module. -/
 @[simp]
 theorem torsion_torsion_eq_top : torsion R (torsion R M) = ⊤ :=
   torsion'_torsion'_eq_top R⁰
 #align submodule.torsion_torsion_eq_top Submodule.torsion_torsion_eq_top
+-/
 
+#print Submodule.torsion_isTorsion /-
 /-- The torsion submodule is always a torsion module. -/
 theorem torsion_isTorsion : Module.IsTorsion R (torsion R M) :=
   torsion'_isTorsion' R⁰
 #align submodule.torsion_is_torsion Submodule.torsion_isTorsion
+-/
 
 end Torsion'
 
@@ -671,13 +741,16 @@ open scoped BigOperators
 
 variable (R M)
 
+#print Module.isTorsionBySet_annihilator_top /-
 theorem Module.isTorsionBySet_annihilator_top :
     Module.IsTorsionBySet R M (⊤ : Submodule R M).annihilator := fun x ha =>
   mem_annihilator.mp ha.Prop x mem_top
 #align module.is_torsion_by_set_annihilator_top Module.isTorsionBySet_annihilator_top
+-/
 
 variable {R M}
 
+#print Submodule.annihilator_top_inter_nonZeroDivisors /-
 theorem Submodule.annihilator_top_inter_nonZeroDivisors [Module.Finite R M]
     (hM : Module.IsTorsion R M) : ((⊤ : Submodule R M).annihilator : Set R) ∩ R⁰ ≠ ∅ :=
   by
@@ -689,9 +762,11 @@ theorem Submodule.annihilator_top_inter_nonZeroDivisors [Module.Finite R M]
   rw [← Finset.prod_erase_mul _ _ n.prop, mul_smul, ← Submonoid.smul_def, (@hM n).choose_spec,
     smul_zero]
 #align submodule.annihilator_top_inter_non_zero_divisors Submodule.annihilator_top_inter_nonZeroDivisors
+-/
 
 variable [NoZeroDivisors R] [Nontrivial R]
 
+#print Submodule.coe_torsion_eq_annihilator_ne_bot /-
 theorem coe_torsion_eq_annihilator_ne_bot :
     (torsion R M : Set M) = {x : M | (R ∙ x).annihilator ≠ ⊥} :=
   by
@@ -702,7 +777,9 @@ theorem coe_torsion_eq_annihilator_ne_bot :
         nonZeroDivisors.coe_ne_zero _⟩,
       fun ⟨a, hax, ha⟩ => ⟨⟨_, mem_nonZeroDivisors_of_ne_zero ha⟩, hax x ⟨1, one_smul _ _⟩⟩⟩
 #align submodule.coe_torsion_eq_annihilator_ne_bot Submodule.coe_torsion_eq_annihilator_ne_bot
+-/
 
+#print Submodule.noZeroSMulDivisors_iff_torsion_eq_bot /-
 /-- A module over a domain has `no_zero_smul_divisors` iff its torsion submodule is trivial. -/
 theorem noZeroSMulDivisors_iff_torsion_eq_bot : NoZeroSMulDivisors R M ↔ torsion R M = ⊥ :=
   by
@@ -722,6 +799,7 @@ theorem noZeroSMulDivisors_iff_torsion_eq_bot : NoZeroSMulDivisors R M ↔ torsi
           · right; rw [← mem_bot _, ← h]
             exact ⟨⟨a, mem_nonZeroDivisors_of_ne_zero ha⟩, hax⟩ }
 #align submodule.no_zero_smul_divisors_iff_torsion_eq_bot Submodule.noZeroSMulDivisors_iff_torsion_eq_bot
+-/
 
 end Torsion
 
@@ -729,6 +807,7 @@ namespace QuotientTorsion
 
 variable [CommRing R] [AddCommGroup M] [Module R M]
 
+#print Submodule.QuotientTorsion.torsion_eq_bot /-
 /-- Quotienting by the torsion submodule gives a torsion-free module. -/
 @[simp]
 theorem torsion_eq_bot : torsion R (M ⧸ torsion R M) = ⊥ :=
@@ -740,10 +819,13 @@ theorem torsion_eq_bot : torsion R (M ⧸ torsion R M) = ⊥ :=
       cases' hax with b h
       exact ⟨b * a, (mul_smul _ _ _).trans h⟩
 #align submodule.quotient_torsion.torsion_eq_bot Submodule.QuotientTorsion.torsion_eq_bot
+-/
 
+#print Submodule.QuotientTorsion.noZeroSMulDivisors /-
 instance noZeroSMulDivisors [IsDomain R] : NoZeroSMulDivisors R (M ⧸ torsion R M) :=
   noZeroSMulDivisors_iff_torsion_eq_bot.mpr torsion_eq_bot
 #align submodule.quotient_torsion.no_zero_smul_divisors Submodule.QuotientTorsion.noZeroSMulDivisors
+-/
 
 end QuotientTorsion
 
@@ -755,6 +837,7 @@ section
 
 variable [Monoid R] [AddCommMonoid M] [DistribMulAction R M]
 
+#print Submodule.isTorsion'_powers_iff /-
 theorem isTorsion'_powers_iff (p : R) :
     IsTorsion' M (Submonoid.powers p) ↔ ∀ x : M, ∃ n : ℕ, p ^ n • x = 0 :=
   ⟨fun h x =>
@@ -764,24 +847,30 @@ theorem isTorsion'_powers_iff (p : R) :
     let ⟨n, hn⟩ := h x
     ⟨⟨_, ⟨n, rfl⟩⟩, hn⟩⟩
 #align submodule.is_torsion'_powers_iff Submodule.isTorsion'_powers_iff
+-/
 
+#print Submodule.pOrder /-
 /-- In a `p ^ ∞`-torsion module (that is, a module where all elements are cancelled by scalar
 multiplication by some power of `p`), the smallest `n` such that `p ^ n • x = 0`.-/
 def pOrder {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (x : M)
     [∀ n : ℕ, Decidable (p ^ n • x = 0)] :=
   Nat.find <| (isTorsion'_powers_iff p).mp hM x
 #align submodule.p_order Submodule.pOrder
+-/
 
+#print Submodule.pow_pOrder_smul /-
 @[simp]
 theorem pow_pOrder_smul {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (x : M)
     [∀ n : ℕ, Decidable (p ^ n • x = 0)] : p ^ pOrder hM x • x = 0 :=
   Nat.find_spec <| (isTorsion'_powers_iff p).mp hM x
 #align submodule.pow_p_order_smul Submodule.pow_pOrder_smul
+-/
 
 end
 
 variable [CommSemiring R] [AddCommMonoid M] [Module R M] [∀ x : M, Decidable (x = 0)]
 
+#print Submodule.exists_isTorsionBy /-
 theorem exists_isTorsionBy {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (d : ℕ) (hd : d ≠ 0)
     (s : Fin d → M) (hs : span R (Set.range s) = ⊤) :
     ∃ j : Fin d, Module.IsTorsionBy R M (p ^ pOrder hM (s j)) :=
@@ -798,6 +887,7 @@ theorem exists_isTorsionBy {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (d
     List.le_of_mem_argmax (List.mem_finRange i) (Option.get_mem hoj)
   rw [← Nat.sub_add_cancel this, pow_add, mul_smul, pow_p_order_smul, smul_zero]
 #align submodule.exists_is_torsion_by Submodule.exists_isTorsionBy
+-/
 
 end PTorsion
 
@@ -807,6 +897,7 @@ namespace Ideal.Quotient
 
 open Submodule
 
+#print Ideal.Quotient.torsionBy_eq_span_singleton /-
 theorem torsionBy_eq_span_singleton {R : Type _} [CommRing R] (a b : R) (ha : a ∈ R⁰) :
     torsionBy R (R ⧸ R ∙ a * b) a = R ∙ mk _ b :=
   by
@@ -821,6 +912,7 @@ theorem torsionBy_eq_span_singleton {R : Type _} [CommRing R] (a b : R) (ha : a
     rw [← h, smul_comm, ← mk_eq_mk, ← quotient.mk_smul,
       (quotient.mk_eq_zero _).mpr <| mem_span_singleton_self _, smul_zero]
 #align ideal.quotient.torsion_by_eq_span_singleton Ideal.Quotient.torsionBy_eq_span_singleton
+-/
 
 end Ideal.Quotient
 
@@ -839,6 +931,7 @@ theorem isTorsion_iff_isTorsion_nat [AddCommMonoid M] :
 #align add_monoid.is_torsion_iff_is_torsion_nat AddMonoid.isTorsion_iff_isTorsion_nat
 -/
 
+#print AddMonoid.isTorsion_iff_isTorsion_int /-
 theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
     AddMonoid.IsTorsion M ↔ Module.IsTorsion ℤ M :=
   by
@@ -851,6 +944,7 @@ theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
     obtain ⟨n, hn⟩ := @h x
     exact exists_nsmul_eq_zero_of_zsmul_eq_zero (nonZeroDivisors.coe_ne_zero n) hn
 #align add_monoid.is_torsion_iff_is_torsion_int AddMonoid.isTorsion_iff_isTorsion_int
+-/
 
 end AddMonoid
 

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

@@ -190,7 +190,7 @@ def torsionBySet (s : Set R) : Submodule R M :=
 @[simps]
 def torsion' (S : Type _) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S R M] :
     Submodule R M where
-  carrier := { x | ∃ a : S, a • x = 0 }
+  carrier := {x | ∃ a : S, a • x = 0}
   zero_mem' := ⟨1, smul_zero _⟩
   add_mem' := fun x y ⟨a, hx⟩ ⟨b, hy⟩ =>
     ⟨b * a, by rw [smul_add, mul_smul, mul_comm, mul_smul, hx, hy, smul_zero, smul_zero, add_zero]⟩
@@ -469,7 +469,8 @@ theorem iSup_torsionBy_eq_torsionBy_prod :
 
 theorem supIndep_torsionBy : S.SupIndep fun i => torsionBy R M <| q i :=
   by
-  convert sup_indep_torsion_by_ideal fun i hi j hj ij =>
+  convert
+    sup_indep_torsion_by_ideal fun i hi j hj ij =>
       (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij
   ext : 1; exact (torsion_by_span_singleton_eq _).symm
 #align submodule.sup_indep_torsion_by Submodule.supIndep_torsionBy
@@ -511,7 +512,8 @@ theorem torsionBy_isInternal {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCo
   by
   rw [← Module.isTorsionBySet_span_singleton_iff, Ideal.submodule_span_eq, ←
     Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf] at hM 
-  convert torsion_by_set_is_internal
+  convert
+    torsion_by_set_is_internal
       (fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij) hM
   ext : 1; exact (torsion_by_span_singleton_eq _).symm
 #align submodule.torsion_by_is_internal Submodule.torsionBy_isInternal
@@ -691,7 +693,7 @@ theorem Submodule.annihilator_top_inter_nonZeroDivisors [Module.Finite R M]
 variable [NoZeroDivisors R] [Nontrivial R]
 
 theorem coe_torsion_eq_annihilator_ne_bot :
-    (torsion R M : Set M) = { x : M | (R ∙ x).annihilator ≠ ⊥ } :=
+    (torsion R M : Set M) = {x : M | (R ∙ x).annihilator ≠ ⊥} :=
   by
   ext x; simp_rw [Submodule.ne_bot_iff, mem_annihilator, mem_span_singleton]
   exact

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

@@ -108,9 +108,9 @@ theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDiv
     torsionOf R M m = ⊥ ↔ m ≠ 0 :=
   by
   refine' ⟨fun h contra => _, fun h => (Submodule.eq_bot_iff _).mpr fun r hr => _⟩
-  · rw [contra, torsion_of_zero] at h
+  · rw [contra, torsion_of_zero] at h 
     exact bot_ne_top.symm h
-  · rw [mem_torsion_of_iff, smul_eq_zero] at hr
+  · rw [mem_torsion_of_iff, smul_eq_zero] at hr 
     tauto
 #align ideal.torsion_of_eq_bot_iff_of_no_zero_smul_divisors Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisors
 
@@ -122,7 +122,7 @@ theorem CompleteLattice.Independent.linear_independent' {ι R M : Type _} {v : 
   by
   refine' linear_independent_iff_not_smul_mem_span.mpr fun i r hi => _
   replace hv := complete_lattice.independent_def.mp hv i
-  simp only [iSup_subtype', ← Submodule.span_range_eq_iSup, disjoint_iff] at hv
+  simp only [iSup_subtype', ← Submodule.span_range_eq_iSup, disjoint_iff] at hv 
   have : r • v i ∈ ⊥ := by
     rw [← hv, Submodule.mem_inf]
     refine' ⟨submodule.mem_span_singleton.mpr ⟨r, rfl⟩, _⟩
@@ -299,7 +299,7 @@ theorem torsionBySet_le_torsionBySet_of_subset {s t : Set R} (st : s ⊆ t) :
 theorem torsionBySet_eq_torsionBySet_span : torsionBySet R M s = torsionBySet R M (Ideal.span s) :=
   by
   refine' le_antisymm (fun x hx => _) (torsion_by_set_le_torsion_by_set_of_subset subset_span)
-  rw [mem_torsion_by_set_iff] at hx⊢
+  rw [mem_torsion_by_set_iff] at hx ⊢
   suffices Ideal.span s ≤ Ideal.torsionOf R M x by rintro ⟨a, ha⟩; exact this ha
   rw [Ideal.span_le]; exact fun a ha => hx ⟨a, ha⟩
 #align submodule.torsion_by_set_eq_torsion_by_span Submodule.torsionBySet_eq_torsionBySet_span
@@ -320,7 +320,7 @@ theorem torsionBy_le_torsionBy_of_dvd (a b : R) (dvd : a ∣ b) : torsionBy R M
 
 @[simp]
 theorem torsionBy_one : torsionBy R M 1 = ⊥ :=
-  eq_bot_iff.mpr fun _ h => by rw [mem_torsion_by_iff, one_smul] at h; exact h
+  eq_bot_iff.mpr fun _ h => by rw [mem_torsion_by_iff, one_smul] at h ; exact h
 #align submodule.torsion_by_one Submodule.torsionBy_one
 
 @[simp]
@@ -428,13 +428,13 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
       (mem_supr_finset_iff_exists_sum _ _).mp
         ((Ideal.eq_top_iff_one _).mp <| (Ideal.iSup_iInf_eq_top_iff_pairwise h _).mpr hp)
     refine' ⟨fun i => ⟨(μ i : R) • x, _⟩, _⟩
-    · rw [mem_torsion_by_set_iff] at hx⊢
+    · rw [mem_torsion_by_set_iff] at hx ⊢
       rintro ⟨a, ha⟩; rw [smul_smul]
       suffices : a * μ i ∈ ⨅ i ∈ S, p i; exact hx ⟨_, this⟩
       rw [mem_infi]; intro j; rw [mem_infi]; intro hj
       by_cases ij : j = i
       · rw [ij]; exact Ideal.mul_mem_right _ _ ha
-      · have := coe_mem (μ i); simp only [mem_infi] at this
+      · have := coe_mem (μ i); simp only [mem_infi] at this 
         exact Ideal.mul_mem_left _ _ (this j hj ij)
     · simp_rw [coe_mk]; rw [← Finset.sum_smul, hμ, one_smul]
 #align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf
@@ -446,7 +446,7 @@ theorem supIndep_torsionBySet_ideal : S.SupIndep fun i => torsionBySet R M <| p
     supr_torsion_by_ideal_eq_torsion_by_infi fun i hi j hj ij => hp (hT hi) (hT hj) ij]
   have :=
     @GaloisConnection.u_inf _ _ (OrderDual.toDual _) (OrderDual.toDual _) _ _ _ _ (torsion_gc R M)
-  dsimp at this⊢
+  dsimp at this ⊢
   rw [← this, Ideal.sup_iInf_eq_top, top_coe, torsion_by_univ]
   intro j hj; apply hp hi (hT hj); rintro rfl; exact hiT hj
 #align submodule.sup_indep_torsion_by_ideal Submodule.supIndep_torsionBySet_ideal
@@ -463,7 +463,7 @@ theorem iSup_torsionBy_eq_torsionBy_prod :
   rw [← torsion_by_span_singleton_eq, Ideal.submodule_span_eq, ←
     Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf, ←
     supr_torsion_by_ideal_eq_torsion_by_infi]
-  · congr ; ext : 1; congr ; ext : 1; exact (torsion_by_span_singleton_eq _).symm
+  · congr; ext : 1; congr; ext : 1; exact (torsion_by_span_singleton_eq _).symm
   · exact fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime _ _).mpr (hq hi hj ij)
 #align submodule.supr_torsion_by_eq_torsion_by_prod Submodule.iSup_torsionBy_eq_torsionBy_prod
 
@@ -510,7 +510,7 @@ theorem torsionBy_isInternal {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCo
     DirectSum.IsInternal fun i : S => torsionBy R M <| q i :=
   by
   rw [← Module.isTorsionBySet_span_singleton_iff, Ideal.submodule_span_eq, ←
-    Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf] at hM
+    Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf] at hM 
   convert torsion_by_set_is_internal
       (fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij) hM
   ext : 1; exact (torsion_by_span_singleton_eq _).symm
@@ -532,7 +532,7 @@ def IsTorsionBySet.hasSMul : SMul (R ⧸ I) M
       by
       show b₁ • x = b₂ • x
       have : (-b₁ + b₂) • x = 0 := @hM x ⟨_, quotient_add_group.left_rel_apply.mp h⟩
-      rw [add_smul, neg_smul, neg_add_eq_zero] at this
+      rw [add_smul, neg_smul, neg_add_eq_zero] at this 
       exact this
 #align module.is_torsion_by_set.has_smul Module.IsTorsionBySet.hasSMul
 -/
@@ -707,7 +707,7 @@ theorem noZeroSMulDivisors_iff_torsion_eq_bot : NoZeroSMulDivisors R M ↔ torsi
   constructor <;> intro h
   · haveI : NoZeroSMulDivisors R M := h
     rw [eq_bot_iff]; rintro x ⟨a, hax⟩
-    change (a : R) • x = 0 at hax
+    change (a : R) • x = 0 at hax 
     cases' eq_zero_or_eq_zero_of_smul_eq_zero hax with h0 h0
     · exfalso; exact nonZeroDivisors.coe_ne_zero a h0; · exact h0
   ·
@@ -733,7 +733,7 @@ theorem torsion_eq_bot : torsion R (M ⧸ torsion R M) = ⊥ :=
   eq_bot_iff.mpr fun z =>
     Quotient.inductionOn' z fun x ⟨a, hax⟩ =>
       by
-      rw [Quotient.mk''_eq_mk', ← quotient.mk_smul, quotient.mk_eq_zero] at hax
+      rw [Quotient.mk''_eq_mk', ← quotient.mk_smul, quotient.mk_eq_zero] at hax 
       rw [mem_bot, Quotient.mk''_eq_mk', quotient.mk_eq_zero]
       cases' hax with b h
       exact ⟨b * a, (mul_smul _ _ _).trans h⟩
@@ -810,10 +810,10 @@ theorem torsionBy_eq_span_singleton {R : Type _} [CommRing R] (a b : R) (ha : a
   by
   ext x; rw [mem_torsion_by_iff, mem_span_singleton]
   obtain ⟨x, rfl⟩ := mk_surjective x; constructor <;> intro h
-  · rw [← mk_eq_mk, ← quotient.mk_smul, quotient.mk_eq_zero, mem_span_singleton] at h
+  · rw [← mk_eq_mk, ← quotient.mk_smul, quotient.mk_eq_zero, mem_span_singleton] at h 
     obtain ⟨c, h⟩ := h;
     rw [smul_eq_mul, smul_eq_mul, mul_comm, mul_assoc, mul_cancel_left_mem_nonZeroDivisors ha,
-      mul_comm] at h
+      mul_comm] at h 
     use c; rw [← h, ← mk_eq_mk, ← quotient.mk_smul, smul_eq_mul, mk_eq_mk]
   · obtain ⟨c, h⟩ := h
     rw [← h, smul_comm, ← mk_eq_mk, ← quotient.mk_smul,

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

@@ -160,7 +160,7 @@ end
 
 end Ideal
 
-open nonZeroDivisors
+open scoped nonZeroDivisors
 
 section Defs
 
@@ -405,7 +405,7 @@ variable {R M}
 
 section Coprime
 
-open BigOperators
+open scoped BigOperators
 
 variable {ι : Type _} {p : ι → Ideal R} {S : Finset ι}
 
@@ -486,7 +486,7 @@ variable [CommRing R] [AddCommGroup M] [Module R M]
 
 namespace Submodule
 
-open BigOperators
+open scoped BigOperators
 
 variable {ι : Type _} [DecidableEq ι] {S : Finset ι}
 
@@ -665,7 +665,7 @@ section Torsion
 
 variable [CommSemiring R] [AddCommMonoid M] [Module R M]
 
-open BigOperators
+open scoped BigOperators
 
 variable (R M)
 

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

@@ -82,24 +82,12 @@ def torsionOf (x : M) : Ideal R :=
 #align ideal.torsion_of Ideal.torsionOf
 -/
 
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 @[simp]
 theorem torsionOf_zero : torsionOf R M (0 : M) = ⊤ := by simp [torsion_of]
 #align ideal.torsion_of_zero Ideal.torsionOf_zero
 
 variable {R M}
 
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 @[simp]
 theorem mem_torsionOf_iff (x : M) (a : R) : a ∈ torsionOf R M x ↔ a • x = 0 :=
   Iff.rfl
@@ -107,12 +95,6 @@ theorem mem_torsionOf_iff (x : M) (a : R) : a ∈ torsionOf R M x ↔ a • x =
 
 variable (R)
 
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 @[simp]
 theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 :=
   by
@@ -121,12 +103,6 @@ theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 :=
   exact Submodule.mem_top
 #align ideal.torsion_of_eq_top_iff Ideal.torsionOf_eq_top_iff
 
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 @[simp]
 theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDivisors R M] (m : M) :
     torsionOf R M m = ⊥ ↔ m ≠ 0 :=
@@ -138,12 +114,6 @@ theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDiv
     tauto
 #align ideal.torsion_of_eq_bot_iff_of_no_zero_smul_divisors Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisors
 
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 /-- See also `complete_lattice.independent.linear_independent` which provides the same conclusion
 but requires the stronger hypothesis `no_zero_smul_divisors R M`. -/
 theorem CompleteLattice.Independent.linear_independent' {ι R M : Type _} {v : ι → M} [Ring R]
@@ -179,9 +149,6 @@ noncomputable def quotTorsionOfEquivSpanSingleton (x : M) : (R ⧸ torsionOf R M
 
 variable {R M}
 
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-<too large>
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 @[simp]
 theorem quotTorsionOfEquivSpanSingleton_apply_mk (x : M) (a : R) :
     quotTorsionOfEquivSpanSingleton R M x (Submodule.Quotient.mk a) =
@@ -218,12 +185,6 @@ def torsionBySet (s : Set R) : Submodule R M :=
 #align submodule.torsion_by_set Submodule.torsionBySet
 -/
 
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 /-- The `S`-torsion submodule, containing all elements `x` of `M` such that `a • x = 0` for some
 `a` in `S`. -/
 @[simps]
@@ -301,34 +262,16 @@ theorem smul_torsionBy (x : torsionBy R M a) : a • x = 0 :=
 #align submodule.smul_torsion_by Submodule.smul_torsionBy
 -/
 
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 @[simp]
 theorem smul_coe_torsionBy (x : torsionBy R M a) : a • (x : M) = 0 :=
   x.Prop
 #align submodule.smul_coe_torsion_by Submodule.smul_coe_torsionBy
 
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 @[simp]
 theorem mem_torsionBy_iff (x : M) : x ∈ torsionBy R M a ↔ a • x = 0 :=
   Iff.rfl
 #align submodule.mem_torsion_by_iff Submodule.mem_torsionBy_iff
 
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 @[simp]
 theorem mem_torsionBySet_iff (x : M) : x ∈ torsionBySet R M s ↔ ∀ a : s, (a : R) • x = 0 :=
   by
@@ -346,12 +289,6 @@ theorem torsionBySet_singleton_eq : torsionBySet R M {a} = torsionBy R M a :=
 #align submodule.torsion_by_singleton_eq Submodule.torsionBySet_singleton_eq
 -/
 
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 theorem torsionBySet_le_torsionBySet_of_subset {s t : Set R} (st : s ⊆ t) :
     torsionBySet R M t ≤ torsionBySet R M s :=
   sInf_le_sInf fun _ ⟨a, ha, h⟩ => ⟨a, st ha, h⟩
@@ -374,12 +311,6 @@ theorem torsionBySet_span_singleton_eq : torsionBySet R M (R ∙ a) = torsionBy
 #align submodule.torsion_by_span_singleton_eq Submodule.torsionBySet_span_singleton_eq
 -/
 
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 theorem torsionBy_le_torsionBy_of_dvd (a b : R) (dvd : a ∣ b) : torsionBy R M a ≤ torsionBy R M b :=
   by
   rw [← torsion_by_span_singleton_eq, ← torsion_by_singleton_eq]
@@ -387,23 +318,11 @@ theorem torsionBy_le_torsionBy_of_dvd (a b : R) (dvd : a ∣ b) : torsionBy R M
   rintro c (rfl : c = b); exact ideal.mem_span_singleton.mpr dvd
 #align submodule.torsion_by_le_torsion_by_of_dvd Submodule.torsionBy_le_torsionBy_of_dvd
 
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 @[simp]
 theorem torsionBy_one : torsionBy R M 1 = ⊥ :=
   eq_bot_iff.mpr fun _ h => by rw [mem_torsion_by_iff, one_smul] at h; exact h
 #align submodule.torsion_by_one Submodule.torsionBy_one
 
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 @[simp]
 theorem torsionBySet_univ : torsionBySet R M Set.univ = ⊥ :=
   by
@@ -417,12 +336,6 @@ open Submodule
 
 namespace Module
 
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 @[simp]
 theorem isTorsionBySet_singleton_iff : IsTorsionBySet R M {a} ↔ IsTorsionBy R M a :=
   by
@@ -430,48 +343,24 @@ theorem isTorsionBySet_singleton_iff : IsTorsionBySet R M {a} ↔ IsTorsionBy R
   rintro ⟨b, rfl : b = a⟩; exact @h _
 #align module.is_torsion_by_singleton_iff Module.isTorsionBySet_singleton_iff
 
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 theorem isTorsionBySet_iff_torsionBySet_eq_top :
     IsTorsionBySet R M s ↔ Submodule.torsionBySet R M s = ⊤ :=
   ⟨fun h => eq_top_iff.mpr fun _ _ => (mem_torsionBySet_iff _ _).mpr <| @h _, fun h x => by
     rw [← mem_torsion_by_set_iff, h]; trivial⟩
 #align module.is_torsion_by_set_iff_torsion_by_set_eq_top Module.isTorsionBySet_iff_torsionBySet_eq_top
 
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 /-- A `a`-torsion module is a module whose `a`-torsion submodule is the full space. -/
 theorem isTorsionBy_iff_torsionBy_eq_top : IsTorsionBy R M a ↔ torsionBy R M a = ⊤ := by
   rw [← torsion_by_singleton_eq, ← is_torsion_by_singleton_iff,
     is_torsion_by_set_iff_torsion_by_set_eq_top]
 #align module.is_torsion_by_iff_torsion_by_eq_top Module.isTorsionBy_iff_torsionBy_eq_top
 
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 theorem isTorsionBySet_iff_is_torsion_by_span :
     IsTorsionBySet R M s ↔ IsTorsionBySet R M (Ideal.span s) := by
   rw [is_torsion_by_set_iff_torsion_by_set_eq_top, is_torsion_by_set_iff_torsion_by_set_eq_top,
     torsion_by_set_eq_torsion_by_span]
 #align module.is_torsion_by_set_iff_is_torsion_by_span Module.isTorsionBySet_iff_is_torsion_by_span
 
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 theorem isTorsionBySet_span_singleton_iff : IsTorsionBySet R M (R ∙ a) ↔ IsTorsionBy R M a :=
   (isTorsionBySet_iff_is_torsion_by_span _).symm.trans <| isTorsionBySet_singleton_iff _
 #align module.is_torsion_by_span_singleton_iff Module.isTorsionBySet_span_singleton_iff
@@ -482,43 +371,19 @@ namespace Submodule
 
 open Module
 
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 theorem torsionBySet_isTorsionBySet : IsTorsionBySet R (torsionBySet R M s) s := fun ⟨x, hx⟩ a =>
   Subtype.ext <| (mem_torsionBySet_iff _ _).mp hx a
 #align submodule.torsion_by_set_is_torsion_by_set Submodule.torsionBySet_isTorsionBySet
 
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 /-- The `a`-torsion submodule is a `a`-torsion module. -/
 theorem torsionBy_isTorsionBy : IsTorsionBy R (torsionBy R M a) a := fun _ => smul_torsionBy _ _
 #align submodule.torsion_by_is_torsion_by Submodule.torsionBy_isTorsionBy
 
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 @[simp]
 theorem torsionBy_torsionBy_eq_top : torsionBy R (torsionBy R M a) a = ⊤ :=
   (isTorsionBy_iff_torsionBy_eq_top a).mp <| torsionBy_isTorsionBy a
 #align submodule.torsion_by_torsion_by_eq_top Submodule.torsionBy_torsionBy_eq_top
 
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-Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_set_torsion_by_set_eq_top Submodule.torsionBySet_torsionBySet_eq_topₓ'. -/
 @[simp]
 theorem torsionBySet_torsionBySet_eq_top : torsionBySet R (torsionBySet R M s) s = ⊤ :=
   (isTorsionBySet_iff_torsionBySet_eq_top s).mp <| torsionBySet_isTorsionBySet s
@@ -548,12 +413,6 @@ variable (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
 
 include hp
 
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-Case conversion may be inaccurate. Consider using '#align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInfₓ'. -/
 theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
     (⨆ i ∈ S, torsionBySet R M <| p i) = torsionBySet R M ↑(⨅ i ∈ S, p i) :=
   by
@@ -580,12 +439,6 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
     · simp_rw [coe_mk]; rw [← Finset.sum_smul, hμ, one_smul]
 #align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf
 
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-Case conversion may be inaccurate. Consider using '#align submodule.sup_indep_torsion_by_ideal Submodule.supIndep_torsionBySet_idealₓ'. -/
 theorem supIndep_torsionBySet_ideal : S.SupIndep fun i => torsionBySet R M <| p i :=
   fun T hT i hi hiT =>
   by
@@ -604,12 +457,6 @@ variable {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCoprime on q))
 
 include hq
 
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-Case conversion may be inaccurate. Consider using '#align submodule.supr_torsion_by_eq_torsion_by_prod Submodule.iSup_torsionBy_eq_torsionBy_prodₓ'. -/
 theorem iSup_torsionBy_eq_torsionBy_prod :
     (⨆ i ∈ S, torsionBy R M <| q i) = torsionBy R M (∏ i in S, q i) :=
   by
@@ -620,12 +467,6 @@ theorem iSup_torsionBy_eq_torsionBy_prod :
   · exact fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime _ _).mpr (hq hi hj ij)
 #align submodule.supr_torsion_by_eq_torsion_by_prod Submodule.iSup_torsionBy_eq_torsionBy_prod
 
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-Case conversion may be inaccurate. Consider using '#align submodule.sup_indep_torsion_by Submodule.supIndep_torsionByₓ'. -/
 theorem supIndep_torsionBy : S.SupIndep fun i => torsionBy R M <| q i :=
   by
   convert sup_indep_torsion_by_ideal fun i hi j hj ij =>
@@ -649,9 +490,6 @@ open BigOperators
 
 variable {ι : Type _} [DecidableEq ι] {S : Finset ι}
 
-/- warning: submodule.torsion_by_set_is_internal -> Submodule.torsionBySet_isInternal is a dubious translation:
-<too large>
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 /-- If the `p i` are pairwise coprime, a `⨅ i, p i`-torsion module is the internal direct sum of
 its `p i`-torsion submodules.-/
 theorem torsionBySet_isInternal {p : ι → Ideal R}
@@ -665,12 +503,6 @@ theorem torsionBySet_isInternal {p : ι → Ideal R}
         (Module.isTorsionBySet_iff_torsionBySet_eq_top _).mp hM)
 #align submodule.torsion_by_set_is_internal Submodule.torsionBySet_isInternal
 
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-Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_is_internal Submodule.torsionBy_isInternalₓ'. -/
 /-- If the `q i` are pairwise coprime, a `∏ i, q i`-torsion module is the internal direct sum of
 its `q i`-torsion submodules.-/
 theorem torsionBy_isInternal {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCoprime on q))
@@ -705,9 +537,6 @@ def IsTorsionBySet.hasSMul : SMul (R ⧸ I) M
 #align module.is_torsion_by_set.has_smul Module.IsTorsionBySet.hasSMul
 -/
 
-/- warning: module.is_torsion_by_set.mk_smul -> Module.IsTorsionBySet.mk_smul is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_set.mk_smul Module.IsTorsionBySet.mk_smulₓ'. -/
 @[simp]
 theorem IsTorsionBySet.mk_smul (b : R) (x : M) :
     haveI := hM.has_smul
@@ -723,9 +552,6 @@ def IsTorsionBySet.module : Module (R ⧸ I) M :=
 #align module.is_torsion_by_set.module Module.IsTorsionBySet.module
 -/
 
-/- warning: module.is_torsion_by_set.is_scalar_tower -> Module.IsTorsionBySet.isScalarTower is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_set.is_scalar_tower Module.IsTorsionBySet.isScalarTowerₓ'. -/
 instance IsTorsionBySet.isScalarTower {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M]
     [IsScalarTower S R R] : @IsScalarTower S (R ⧸ I) M _ (IsTorsionBySet.module hM).toSMul _
     where smul_assoc b d x := Quotient.inductionOn' d fun c => (smul_assoc b c x : _)
@@ -747,9 +573,6 @@ namespace Submodule
 instance (I : Ideal R) : Module (R ⧸ I) (torsionBySet R M I) :=
   Module.IsTorsionBySet.module <| torsionBySet_isTorsionBySet I
 
-/- warning: submodule.torsion_by_set.mk_smul -> Submodule.torsionBySet.mk_smul is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_set.mk_smul Submodule.torsionBySet.mk_smulₓ'. -/
 @[simp]
 theorem torsionBySet.mk_smul (I : Ideal R) (b : R) (x : torsionBySet R M I) :
     Ideal.Quotient.mk I b • x = b • x :=
@@ -765,9 +588,6 @@ instance (a : R) : Module (R ⧸ R ∙ a) (torsionBy R M a) :=
   Module.IsTorsionBySet.module <|
     (Module.isTorsionBySet_span_singleton_iff a).mpr <| torsionBy_isTorsionBy a
 
-/- warning: submodule.torsion_by.mk_smul -> Submodule.torsionBy.mk_smul is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.torsion_by.mk_smul Submodule.torsionBy.mk_smulₓ'. -/
 @[simp]
 theorem torsionBy.mk_smul (a b : R) (x : torsionBy R M a) :
     Ideal.Quotient.mk (R ∙ a) b • x = b • x :=
@@ -792,20 +612,11 @@ variable [CommSemiring R] [AddCommMonoid M] [Module R M]
 
 variable (S : Type _) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S R M]
 
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 @[simp]
 theorem mem_torsion'_iff (x : M) : x ∈ torsion' R M S ↔ ∃ a : S, a • x = 0 :=
   Iff.rfl
 #align submodule.mem_torsion'_iff Submodule.mem_torsion'_iff
 
-/- warning: submodule.mem_torsion_iff -> Submodule.mem_torsion_iff is a dubious translation:
-<too large>
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 @[simp]
 theorem mem_torsion_iff (x : M) : x ∈ torsion R M ↔ ∃ a : R⁰, a • x = 0 :=
   Iff.rfl
@@ -822,41 +633,20 @@ instance : DistribMulAction S (torsion' R M S) :=
 instance : SMulCommClass S R (torsion' R M S) :=
   ⟨fun s a x => Subtype.ext <| smul_comm _ _ _⟩
 
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 /-- A `S`-torsion module is a module whose `S`-torsion submodule is the full space. -/
 theorem isTorsion'_iff_torsion'_eq_top : IsTorsion' M S ↔ torsion' R M S = ⊤ :=
   ⟨fun h => eq_top_iff.mpr fun _ _ => @h _, fun h x => by rw [← @mem_torsion'_iff R, h]; trivial⟩
 #align submodule.is_torsion'_iff_torsion'_eq_top Submodule.isTorsion'_iff_torsion'_eq_top
 
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 /-- The `S`-torsion submodule is a `S`-torsion module. -/
 theorem torsion'_isTorsion' : IsTorsion' (torsion' R M S) S := fun ⟨x, ⟨a, h⟩⟩ => ⟨a, Subtype.ext h⟩
 #align submodule.torsion'_is_torsion' Submodule.torsion'_isTorsion'
 
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 @[simp]
 theorem torsion'_torsion'_eq_top : torsion' R (torsion' R M S) S = ⊤ :=
   (isTorsion'_iff_torsion'_eq_top S).mp <| torsion'_isTorsion' S
 #align submodule.torsion'_torsion'_eq_top Submodule.torsion'_torsion'_eq_top
 
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 /-- The torsion submodule of the torsion submodule (viewed as a module) is the full
 torsion module. -/
 @[simp]
@@ -864,12 +654,6 @@ theorem torsion_torsion_eq_top : torsion R (torsion R M) = ⊤ :=
   torsion'_torsion'_eq_top R⁰
 #align submodule.torsion_torsion_eq_top Submodule.torsion_torsion_eq_top
 
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 /-- The torsion submodule is always a torsion module. -/
 theorem torsion_isTorsion : Module.IsTorsion R (torsion R M) :=
   torsion'_isTorsion' R⁰
@@ -885,12 +669,6 @@ open BigOperators
 
 variable (R M)
 
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 theorem Module.isTorsionBySet_annihilator_top :
     Module.IsTorsionBySet R M (⊤ : Submodule R M).annihilator := fun x ha =>
   mem_annihilator.mp ha.Prop x mem_top
@@ -898,12 +676,6 @@ theorem Module.isTorsionBySet_annihilator_top :
 
 variable {R M}
 
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 theorem Submodule.annihilator_top_inter_nonZeroDivisors [Module.Finite R M]
     (hM : Module.IsTorsion R M) : ((⊤ : Submodule R M).annihilator : Set R) ∩ R⁰ ≠ ∅ :=
   by
@@ -918,12 +690,6 @@ theorem Submodule.annihilator_top_inter_nonZeroDivisors [Module.Finite R M]
 
 variable [NoZeroDivisors R] [Nontrivial R]
 
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 theorem coe_torsion_eq_annihilator_ne_bot :
     (torsion R M : Set M) = { x : M | (R ∙ x).annihilator ≠ ⊥ } :=
   by
@@ -935,12 +701,6 @@ theorem coe_torsion_eq_annihilator_ne_bot :
       fun ⟨a, hax, ha⟩ => ⟨⟨_, mem_nonZeroDivisors_of_ne_zero ha⟩, hax x ⟨1, one_smul _ _⟩⟩⟩
 #align submodule.coe_torsion_eq_annihilator_ne_bot Submodule.coe_torsion_eq_annihilator_ne_bot
 
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 /-- A module over a domain has `no_zero_smul_divisors` iff its torsion submodule is trivial. -/
 theorem noZeroSMulDivisors_iff_torsion_eq_bot : NoZeroSMulDivisors R M ↔ torsion R M = ⊥ :=
   by
@@ -967,9 +727,6 @@ namespace QuotientTorsion
 
 variable [CommRing R] [AddCommGroup M] [Module R M]
 
-/- warning: submodule.quotient_torsion.torsion_eq_bot -> Submodule.QuotientTorsion.torsion_eq_bot is a dubious translation:
-<too large>
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 /-- Quotienting by the torsion submodule gives a torsion-free module. -/
 @[simp]
 theorem torsion_eq_bot : torsion R (M ⧸ torsion R M) = ⊥ :=
@@ -982,9 +739,6 @@ theorem torsion_eq_bot : torsion R (M ⧸ torsion R M) = ⊥ :=
       exact ⟨b * a, (mul_smul _ _ _).trans h⟩
 #align submodule.quotient_torsion.torsion_eq_bot Submodule.QuotientTorsion.torsion_eq_bot
 
-/- warning: submodule.quotient_torsion.no_zero_smul_divisors -> Submodule.QuotientTorsion.noZeroSMulDivisors is a dubious translation:
-<too large>
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 instance noZeroSMulDivisors [IsDomain R] : NoZeroSMulDivisors R (M ⧸ torsion R M) :=
   noZeroSMulDivisors_iff_torsion_eq_bot.mpr torsion_eq_bot
 #align submodule.quotient_torsion.no_zero_smul_divisors Submodule.QuotientTorsion.noZeroSMulDivisors
@@ -999,12 +753,6 @@ section
 
 variable [Monoid R] [AddCommMonoid M] [DistribMulAction R M]
 
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 theorem isTorsion'_powers_iff (p : R) :
     IsTorsion' M (Submonoid.powers p) ↔ ∀ x : M, ∃ n : ℕ, p ^ n • x = 0 :=
   ⟨fun h x =>
@@ -1015,12 +763,6 @@ theorem isTorsion'_powers_iff (p : R) :
     ⟨⟨_, ⟨n, rfl⟩⟩, hn⟩⟩
 #align submodule.is_torsion'_powers_iff Submodule.isTorsion'_powers_iff
 
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-Case conversion may be inaccurate. Consider using '#align submodule.p_order Submodule.pOrderₓ'. -/
 /-- In a `p ^ ∞`-torsion module (that is, a module where all elements are cancelled by scalar
 multiplication by some power of `p`), the smallest `n` such that `p ^ n • x = 0`.-/
 def pOrder {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (x : M)
@@ -1028,12 +770,6 @@ def pOrder {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (x : M)
   Nat.find <| (isTorsion'_powers_iff p).mp hM x
 #align submodule.p_order Submodule.pOrder
 
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-Case conversion may be inaccurate. Consider using '#align submodule.pow_p_order_smul Submodule.pow_pOrder_smulₓ'. -/
 @[simp]
 theorem pow_pOrder_smul {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (x : M)
     [∀ n : ℕ, Decidable (p ^ n • x = 0)] : p ^ pOrder hM x • x = 0 :=
@@ -1044,9 +780,6 @@ end
 
 variable [CommSemiring R] [AddCommMonoid M] [Module R M] [∀ x : M, Decidable (x = 0)]
 
-/- warning: submodule.exists_is_torsion_by -> Submodule.exists_isTorsionBy is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.exists_is_torsion_by Submodule.exists_isTorsionByₓ'. -/
 theorem exists_isTorsionBy {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (d : ℕ) (hd : d ≠ 0)
     (s : Fin d → M) (hs : span R (Set.range s) = ⊤) :
     ∃ j : Fin d, Module.IsTorsionBy R M (p ^ pOrder hM (s j)) :=
@@ -1072,9 +805,6 @@ namespace Ideal.Quotient
 
 open Submodule
 
-/- warning: ideal.quotient.torsion_by_eq_span_singleton -> Ideal.Quotient.torsionBy_eq_span_singleton is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align ideal.quotient.torsion_by_eq_span_singleton Ideal.Quotient.torsionBy_eq_span_singletonₓ'. -/
 theorem torsionBy_eq_span_singleton {R : Type _} [CommRing R] (a b : R) (ha : a ∈ R⁰) :
     torsionBy R (R ⧸ R ∙ a * b) a = R ∙ mk _ b :=
   by
@@ -1107,12 +837,6 @@ theorem isTorsion_iff_isTorsion_nat [AddCommMonoid M] :
 #align add_monoid.is_torsion_iff_is_torsion_nat AddMonoid.isTorsion_iff_isTorsion_nat
 -/
 
-/- warning: add_monoid.is_torsion_iff_is_torsion_int -> AddMonoid.isTorsion_iff_isTorsion_int is a dubious translation:
-lean 3 declaration is
-  forall {M : Type.{u1}} [_inst_1 : AddCommGroup.{u1} M], Iff (AddMonoid.IsTorsion.{u1} M (SubNegMonoid.toAddMonoid.{u1} M (AddGroup.toSubNegMonoid.{u1} M (AddCommGroup.toAddGroup.{u1} M _inst_1)))) (Module.IsTorsion.{0, u1} Int M Int.commSemiring (AddCommGroup.toAddCommMonoid.{u1} M _inst_1) (AddCommGroup.intModule.{u1} M _inst_1))
-but is expected to have type
-  forall {M : Type.{u1}} [_inst_1 : AddCommGroup.{u1} M], Iff (AddMonoid.IsTorsion.{u1} M (SubNegMonoid.toAddMonoid.{u1} M (AddGroup.toSubNegMonoid.{u1} M (AddCommGroup.toAddGroup.{u1} M _inst_1)))) (Module.IsTorsion.{0, u1} Int M Int.instCommSemiringInt (AddCommGroup.toAddCommMonoid.{u1} M _inst_1) (AddCommGroup.intModule.{u1} M _inst_1))
-Case conversion may be inaccurate. Consider using '#align add_monoid.is_torsion_iff_is_torsion_int AddMonoid.isTorsion_iff_isTorsion_intₓ'. -/
 theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
     AddMonoid.IsTorsion M ↔ Module.IsTorsion ℤ M :=
   by

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

@@ -363,12 +363,8 @@ theorem torsionBySet_eq_torsionBySet_span : torsionBySet R M s = torsionBySet R
   by
   refine' le_antisymm (fun x hx => _) (torsion_by_set_le_torsion_by_set_of_subset subset_span)
   rw [mem_torsion_by_set_iff] at hx⊢
-  suffices Ideal.span s ≤ Ideal.torsionOf R M x
-    by
-    rintro ⟨a, ha⟩
-    exact this ha
-  rw [Ideal.span_le]
-  exact fun a ha => hx ⟨a, ha⟩
+  suffices Ideal.span s ≤ Ideal.torsionOf R M x by rintro ⟨a, ha⟩; exact this ha
+  rw [Ideal.span_le]; exact fun a ha => hx ⟨a, ha⟩
 #align submodule.torsion_by_set_eq_torsion_by_span Submodule.torsionBySet_eq_torsionBySet_span
 -/
 
@@ -399,9 +395,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_one Submodule.torsionBy_oneₓ'. -/
 @[simp]
 theorem torsionBy_one : torsionBy R M 1 = ⊥ :=
-  eq_bot_iff.mpr fun _ h => by
-    rw [mem_torsion_by_iff, one_smul] at h
-    exact h
+  eq_bot_iff.mpr fun _ h => by rw [mem_torsion_by_iff, one_smul] at h; exact h
 #align submodule.torsion_by_one Submodule.torsionBy_one
 
 /- warning: submodule.torsion_by_univ -> Submodule.torsionBySet_univ is a dubious translation:
@@ -444,10 +438,8 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_set_iff_torsion_by_set_eq_top Module.isTorsionBySet_iff_torsionBySet_eq_topₓ'. -/
 theorem isTorsionBySet_iff_torsionBySet_eq_top :
     IsTorsionBySet R M s ↔ Submodule.torsionBySet R M s = ⊤ :=
-  ⟨fun h => eq_top_iff.mpr fun _ _ => (mem_torsionBySet_iff _ _).mpr <| @h _, fun h x =>
-    by
-    rw [← mem_torsion_by_set_iff, h]
-    trivial⟩
+  ⟨fun h => eq_top_iff.mpr fun _ _ => (mem_torsionBySet_iff _ _).mpr <| @h _, fun h x => by
+    rw [← mem_torsion_by_set_iff, h]; trivial⟩
 #align module.is_torsion_by_set_iff_torsion_by_set_eq_top Module.isTorsionBySet_iff_torsionBySet_eq_top
 
 /- warning: module.is_torsion_by_iff_torsion_by_eq_top -> Module.isTorsionBy_iff_torsionBy_eq_top is a dubious translation:
@@ -566,16 +558,9 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
     (⨆ i ∈ S, torsionBySet R M <| p i) = torsionBySet R M ↑(⨅ i ∈ S, p i) :=
   by
   cases' S.eq_empty_or_nonempty with h h
-  · rw [h]
-    convert iSup_emptyset
-    convert torsion_by_univ
-    convert top_coe
-    exact iInf_emptyset
+  · rw [h]; convert iSup_emptyset; convert torsion_by_univ; convert top_coe; exact iInf_emptyset
   apply le_antisymm
-  · apply iSup_le _
-    intro i
-    apply iSup_le _
-    intro is
+  · apply iSup_le _; intro i; apply iSup_le _; intro is
     apply torsion_by_set_le_torsion_by_set_of_subset
     exact (iInf_le (fun i => ⨅ H : i ∈ S, p i) i).trans (iInf_le _ is)
   · intro x hx
@@ -585,22 +570,14 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
         ((Ideal.eq_top_iff_one _).mp <| (Ideal.iSup_iInf_eq_top_iff_pairwise h _).mpr hp)
     refine' ⟨fun i => ⟨(μ i : R) • x, _⟩, _⟩
     · rw [mem_torsion_by_set_iff] at hx⊢
-      rintro ⟨a, ha⟩
-      rw [smul_smul]
-      suffices : a * μ i ∈ ⨅ i ∈ S, p i
-      exact hx ⟨_, this⟩
-      rw [mem_infi]
-      intro j
-      rw [mem_infi]
-      intro hj
+      rintro ⟨a, ha⟩; rw [smul_smul]
+      suffices : a * μ i ∈ ⨅ i ∈ S, p i; exact hx ⟨_, this⟩
+      rw [mem_infi]; intro j; rw [mem_infi]; intro hj
       by_cases ij : j = i
-      · rw [ij]
-        exact Ideal.mul_mem_right _ _ ha
-      · have := coe_mem (μ i)
-        simp only [mem_infi] at this
+      · rw [ij]; exact Ideal.mul_mem_right _ _ ha
+      · have := coe_mem (μ i); simp only [mem_infi] at this
         exact Ideal.mul_mem_left _ _ (this j hj ij)
-    · simp_rw [coe_mk]
-      rw [← Finset.sum_smul, hμ, one_smul]
+    · simp_rw [coe_mk]; rw [← Finset.sum_smul, hμ, one_smul]
 #align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf
 
 /- warning: submodule.sup_indep_torsion_by_ideal -> Submodule.supIndep_torsionBySet_ideal is a dubious translation:
@@ -639,11 +616,7 @@ theorem iSup_torsionBy_eq_torsionBy_prod :
   rw [← torsion_by_span_singleton_eq, Ideal.submodule_span_eq, ←
     Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf, ←
     supr_torsion_by_ideal_eq_torsion_by_infi]
-  · congr
-    ext : 1
-    congr
-    ext : 1
-    exact (torsion_by_span_singleton_eq _).symm
+  · congr ; ext : 1; congr ; ext : 1; exact (torsion_by_span_singleton_eq _).symm
   · exact fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime _ _).mpr (hq hi hj ij)
 #align submodule.supr_torsion_by_eq_torsion_by_prod Submodule.iSup_torsionBy_eq_torsionBy_prod
 
@@ -840,12 +813,7 @@ theorem mem_torsion_iff (x : M) : x ∈ torsion R M ↔ ∃ a : R⁰, a • x =
 
 @[simps]
 instance : SMul S (torsion' R M S) :=
-  ⟨fun s x =>
-    ⟨s • x, by
-      obtain ⟨x, a, h⟩ := x
-      use a
-      dsimp
-      rw [smul_comm, h, smul_zero]⟩⟩
+  ⟨fun s x => ⟨s • x, by obtain ⟨x, a, h⟩ := x; use a; dsimp; rw [smul_comm, h, smul_zero]⟩⟩
 
 instance : DistribMulAction S (torsion' R M S) :=
   Subtype.coe_injective.DistribMulAction (torsion' R M S).Subtype.toAddMonoidHom fun (c : S) x =>
@@ -862,10 +830,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align submodule.is_torsion'_iff_torsion'_eq_top Submodule.isTorsion'_iff_torsion'_eq_topₓ'. -/
 /-- A `S`-torsion module is a module whose `S`-torsion submodule is the full space. -/
 theorem isTorsion'_iff_torsion'_eq_top : IsTorsion' M S ↔ torsion' R M S = ⊤ :=
-  ⟨fun h => eq_top_iff.mpr fun _ _ => @h _, fun h x =>
-    by
-    rw [← @mem_torsion'_iff R, h]
-    trivial⟩
+  ⟨fun h => eq_top_iff.mpr fun _ _ => @h _, fun h x => by rw [← @mem_torsion'_iff R, h]; trivial⟩
 #align submodule.is_torsion'_iff_torsion'_eq_top Submodule.isTorsion'_iff_torsion'_eq_top
 
 /- warning: submodule.torsion'_is_torsion' -> Submodule.torsion'_isTorsion' is a dubious translation:
@@ -981,23 +946,18 @@ theorem noZeroSMulDivisors_iff_torsion_eq_bot : NoZeroSMulDivisors R M ↔ torsi
   by
   constructor <;> intro h
   · haveI : NoZeroSMulDivisors R M := h
-    rw [eq_bot_iff]
-    rintro x ⟨a, hax⟩
+    rw [eq_bot_iff]; rintro x ⟨a, hax⟩
     change (a : R) • x = 0 at hax
     cases' eq_zero_or_eq_zero_of_smul_eq_zero hax with h0 h0
-    · exfalso
-      exact nonZeroDivisors.coe_ne_zero a h0
-    · exact h0
+    · exfalso; exact nonZeroDivisors.coe_ne_zero a h0; · exact h0
   ·
     exact
       {
         eq_zero_or_eq_zero_of_smul_eq_zero := fun a x hax =>
           by
           by_cases ha : a = 0
-          · left
-            exact ha
-          · right
-            rw [← mem_bot _, ← h]
+          · left; exact ha
+          · right; rw [← mem_bot _, ← h]
             exact ⟨⟨a, mem_nonZeroDivisors_of_ne_zero ha⟩, hax⟩ }
 #align submodule.no_zero_smul_divisors_iff_torsion_eq_bot Submodule.noZeroSMulDivisors_iff_torsion_eq_bot
 
@@ -1121,11 +1081,10 @@ theorem torsionBy_eq_span_singleton {R : Type _} [CommRing R] (a b : R) (ha : a
   ext x; rw [mem_torsion_by_iff, mem_span_singleton]
   obtain ⟨x, rfl⟩ := mk_surjective x; constructor <;> intro h
   · rw [← mk_eq_mk, ← quotient.mk_smul, quotient.mk_eq_zero, mem_span_singleton] at h
-    obtain ⟨c, h⟩ := h
+    obtain ⟨c, h⟩ := h;
     rw [smul_eq_mul, smul_eq_mul, mul_comm, mul_assoc, mul_cancel_left_mem_nonZeroDivisors ha,
       mul_comm] at h
-    use c
-    rw [← h, ← mk_eq_mk, ← quotient.mk_smul, smul_eq_mul, mk_eq_mk]
+    use c; rw [← h, ← mk_eq_mk, ← quotient.mk_smul, smul_eq_mul, mk_eq_mk]
   · obtain ⟨c, h⟩ := h
     rw [← h, smul_comm, ← mk_eq_mk, ← quotient.mk_smul,
       (quotient.mk_eq_zero _).mpr <| mem_span_singleton_self _, smul_zero]

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Pierre-Alexandre Bazin
 
 ! This file was ported from Lean 3 source module algebra.module.torsion
-! leanprover-community/mathlib commit cdc34484a07418af43daf8198beaf5c00324bca8
+! leanprover-community/mathlib commit 0b7c740e25651db0ba63648fbae9f9d6f941e31b
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -18,6 +18,9 @@ import Mathbin.RingTheory.Finiteness
 /-!
 # Torsion submodules
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 ## Main definitions
 
 * `torsion_of R M x` : the torsion ideal of `x`, containing all `a` such that `a • x = 0`.
@@ -177,10 +180,7 @@ noncomputable def quotTorsionOfEquivSpanSingleton (x : M) : (R ⧸ torsionOf R M
 variable {R M}
 
 /- warning: ideal.quot_torsion_of_equiv_span_singleton_apply_mk -> Ideal.quotTorsionOfEquivSpanSingleton_apply_mk is a dubious translation:
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_inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddMonoid.toAddZeroClass.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddCommMonoid.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.Quotient.addCommGroup.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x))))) (DistribMulAction.toDistribSMul.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.Quotient.addCommGroup.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)))) (Module.toDistribMulAction.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.Quotient.addCommGroup.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x))) (Submodule.Quotient.module.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)))))) (SMulZeroClass.toSMul.{u1, u2} R (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (AddMonoid.toZero.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (AddCommMonoid.toAddMonoid.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))))) (DistribSMul.toSMulZeroClass.{u1, u2} R (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (AddMonoid.toAddZeroClass.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (AddCommMonoid.toAddMonoid.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))))) (DistribMulAction.toDistribSMul.{u1, u2} R (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (Module.toDistribMulAction.{u1, u2} R (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (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 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))))))) (DistribMulActionHomClass.toSMulHomClass.{max u1 u2, u1, u1, u2} (LinearEquiv.{u1, u1, u1, u2} 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))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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 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(Set.instSingletonSet.{u2} M) x))) x (Submodule.mem_span_singleton_self.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)))
+<too large>
 Case conversion may be inaccurate. Consider using '#align ideal.quot_torsion_of_equiv_span_singleton_apply_mk Ideal.quotTorsionOfEquivSpanSingleton_apply_mkₓ'. -/
 @[simp]
 theorem quotTorsionOfEquivSpanSingleton_apply_mk (x : M) (a : R) :
@@ -677,10 +677,7 @@ open BigOperators
 variable {ι : Type _} [DecidableEq ι] {S : Finset ι}
 
 /- warning: submodule.torsion_by_set_is_internal -> Submodule.torsionBySet_isInternal is a dubious translation:
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(SetLike.coe.{u3, u3} (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) R (Submodule.setLike.{u3, u3} R R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)))) (p (Subtype.val.{succ u2} ι (fun (x : ι) => Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) x S) i)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_set_is_internal Submodule.torsionBySet_isInternalₓ'. -/
 /-- If the `p i` are pairwise coprime, a `⨅ i, p i`-torsion module is the internal direct sum of
 its `p i`-torsion submodules.-/
@@ -736,10 +733,7 @@ def IsTorsionBySet.hasSMul : SMul (R ⧸ I) M
 -/
 
 /- warning: module.is_torsion_by_set.mk_smul -> Module.IsTorsionBySet.mk_smul is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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(CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))) (RingHom.instRingHomClassRingHom.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I)))))))) (Ideal.Quotient.mk.{u1} R _inst_1 I) b) x) (HSMul.hSMul.{u1, u2, u2} R M M (instHSMul.{u1, u2} R M (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_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{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_2))))) (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_2))))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) b x)
+<too large>
 Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_set.mk_smul Module.IsTorsionBySet.mk_smulₓ'. -/
 @[simp]
 theorem IsTorsionBySet.mk_smul (b : R) (x : M) :
@@ -757,10 +751,7 @@ def IsTorsionBySet.module : Module (R ⧸ I) M :=
 -/
 
 /- warning: module.is_torsion_by_set.is_scalar_tower -> Module.IsTorsionBySet.isScalarTower is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} (hM : Module.IsTorsionBySet.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{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)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) I)) {S : Type.{u3}} [_inst_4 : SMul.{u3, u1} S R] [_inst_5 : SMul.{u3, u2} S M] [_inst_6 : IsScalarTower.{u3, u1, u2} S R 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_2)))) (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_2)))) (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_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) _inst_5] [_inst_7 : IsScalarTower.{u3, u1, u1} S R R _inst_4 (Mul.toSMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) _inst_4], IsScalarTower.{u3, u1, u2} S (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) M (Submodule.Quotient.hasSmul'.{u1, u1, u3} 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))) S _inst_4 _inst_4 _inst_7 I) (MulAction.toHasSmul.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) M (MonoidWithZero.toMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Semiring.toMonoidWithZero.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (DistribMulAction.toMulAction.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) M (MonoidWithZero.toMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Semiring.toMonoidWithZero.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)) (Module.toDistribMulAction.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) M (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (Module.IsTorsionBySet.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 I hM)))) _inst_5
-but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} (hM : Module.IsTorsionBySet.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I)) {S : Type.{u3}} [_inst_4 : SMul.{u3, u1} S R] [_inst_5 : SMul.{u3, u2} S M] [_inst_6 : IsScalarTower.{u3, u1, u2} S R 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_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{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_2))))) (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_2))))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) _inst_5] [_inst_7 : IsScalarTower.{u3, u1, u1} S R R _inst_4 (Algebra.toSMul.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _inst_4], IsScalarTower.{u3, u1, u2} S (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) M (Submodule.Quotient.hasSmul'.{u1, u1, u3} R R (CommRing.toRing.{u1} R _inst_1) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) S _inst_4 _inst_4 _inst_7 I) (MulAction.toSMul.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) M (MonoidWithZero.toMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Semiring.toMonoidWithZero.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) 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(CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (Module.IsTorsionBySet.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 I hM)))) _inst_5
+<too large>
 Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_set.is_scalar_tower Module.IsTorsionBySet.isScalarTowerₓ'. -/
 instance IsTorsionBySet.isScalarTower {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M]
     [IsScalarTower S R R] : @IsScalarTower S (R ⧸ I) M _ (IsTorsionBySet.module hM).toSMul _
@@ -784,10 +775,7 @@ instance (I : Ideal R) : Module (R ⧸ I) (torsionBySet R M I) :=
   Module.IsTorsionBySet.module <| torsionBySet_isTorsionBySet I
 
 /- warning: submodule.torsion_by_set.mk_smul -> Submodule.torsionBySet.mk_smul is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_set.mk_smul Submodule.torsionBySet.mk_smulₓ'. -/
 @[simp]
 theorem torsionBySet.mk_smul (I : Ideal R) (b : R) (x : torsionBySet R M I) :
@@ -805,10 +793,7 @@ instance (a : R) : Module (R ⧸ R ∙ a) (torsionBy R M a) :=
     (Module.isTorsionBySet_span_singleton_iff a).mpr <| torsionBy_isTorsionBy a
 
 /- warning: submodule.torsion_by.mk_smul -> Submodule.torsionBy.mk_smul is a dubious translation:
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(NonUnitalRingHomClass.toMulHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (Ideal.Quotient.commRing.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))))))) R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) 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+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.torsion_by.mk_smul Submodule.torsionBy.mk_smulₓ'. -/
 @[simp]
 theorem torsionBy.mk_smul (a b : R) (x : torsionBy R M a) :
@@ -846,10 +831,7 @@ theorem mem_torsion'_iff (x : M) : x ∈ torsion' R M S ↔ ∃ a : S, a • x =
 #align submodule.mem_torsion'_iff Submodule.mem_torsion'_iff
 
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 Case conversion may be inaccurate. Consider using '#align submodule.mem_torsion_iff Submodule.mem_torsion_iffₓ'. -/
 @[simp]
 theorem mem_torsion_iff (x : M) : x ∈ torsion R M ↔ ∃ a : R⁰, a • x = 0 :=
@@ -897,10 +879,7 @@ theorem torsion'_isTorsion' : IsTorsion' (torsion' R M S) S := fun ⟨x, ⟨a, h
 #align submodule.torsion'_is_torsion' Submodule.torsion'_isTorsion'
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.torsion'_torsion'_eq_top Submodule.torsion'_torsion'_eq_topₓ'. -/
 @[simp]
 theorem torsion'_torsion'_eq_top : torsion' R (torsion' R M S) S = ⊤ :=
@@ -1029,10 +1008,7 @@ namespace QuotientTorsion
 variable [CommRing R] [AddCommGroup M] [Module R M]
 
 /- warning: submodule.quotient_torsion.torsion_eq_bot -> Submodule.QuotientTorsion.torsion_eq_bot is a dubious translation:
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(HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.Quotient.module.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.quotient_torsion.torsion_eq_bot Submodule.QuotientTorsion.torsion_eq_botₓ'. -/
 /-- Quotienting by the torsion submodule gives a torsion-free module. -/
 @[simp]
@@ -1047,10 +1023,7 @@ theorem torsion_eq_bot : torsion R (M ⧸ torsion R M) = ⊥ :=
 #align submodule.quotient_torsion.torsion_eq_bot Submodule.QuotientTorsion.torsion_eq_bot
 
 /- warning: submodule.quotient_torsion.no_zero_smul_divisors -> Submodule.QuotientTorsion.noZeroSMulDivisors is a dubious translation:
-lean 3 declaration is
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_inst_3))))))
-but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_4 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))], NoZeroSMulDivisors.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_4))) (Submodule.Quotient.instZeroQuotientSubmoduleToSemiringToAddCommMonoidHasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.hasSmul.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))
+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.quotient_torsion.no_zero_smul_divisors Submodule.QuotientTorsion.noZeroSMulDivisorsₓ'. -/
 instance noZeroSMulDivisors [IsDomain R] : NoZeroSMulDivisors R (M ⧸ torsion R M) :=
   noZeroSMulDivisors_iff_torsion_eq_bot.mpr torsion_eq_bot
@@ -1112,10 +1085,7 @@ end
 variable [CommSemiring R] [AddCommMonoid M] [Module R M] [∀ x : M, Decidable (x = 0)]
 
 /- warning: submodule.exists_is_torsion_by -> Submodule.exists_isTorsionBy is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.exists_is_torsion_by Submodule.exists_isTorsionByₓ'. -/
 theorem exists_isTorsionBy {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (d : ℕ) (hd : d ≠ 0)
     (s : Fin d → M) (hs : span R (Set.range s) = ⊤) :
@@ -1143,10 +1113,7 @@ namespace Ideal.Quotient
 open Submodule
 
 /- warning: ideal.quotient.torsion_by_eq_span_singleton -> Ideal.Quotient.torsionBy_eq_span_singleton is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align ideal.quotient.torsion_by_eq_span_singleton Ideal.Quotient.torsionBy_eq_span_singletonₓ'. -/
 theorem torsionBy_eq_span_singleton {R : Type _} [CommRing R] (a b : R) (ha : a ∈ R⁰) :
     torsionBy R (R ⧸ R ∙ a * b) a = R ∙ mk _ b :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/e1a18cad9cd462973d760af7de36b05776b8811c

@@ -71,18 +71,32 @@ section TorsionOf
 
 variable (R M : Type _) [Semiring R] [AddCommMonoid M] [Module R M]
 
+#print Ideal.torsionOf /-
 /-- The torsion ideal of `x`, containing all `a` such that `a • x = 0`.-/
 @[simps]
 def torsionOf (x : M) : Ideal R :=
   (LinearMap.toSpanSingleton R M x).ker
 #align ideal.torsion_of Ideal.torsionOf
+-/
 
+/- warning: ideal.torsion_of_zero -> Ideal.torsionOf_zero is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) (M : Type.{u2}) [_inst_1 : Semiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M _inst_1 _inst_2], Eq.{succ u1} (Ideal.{u1} R _inst_1) (Ideal.torsionOf.{u1, u2} R M _inst_1 _inst_2 _inst_3 (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))))))) (Top.top.{u1} (Ideal.{u1} R _inst_1) (Submodule.hasTop.{u1, u1} R R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Semiring.toModule.{u1} R _inst_1)))
+but is expected to have type
+  forall (R : Type.{u2}) (M : Type.{u1}) [_inst_1 : Semiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M _inst_1 _inst_2], Eq.{succ u2} (Ideal.{u2} R _inst_1) (Ideal.torsionOf.{u2, u1} R M _inst_1 _inst_2 _inst_3 (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2))))) (Top.top.{u2} (Ideal.{u2} R _inst_1) (Submodule.instTopSubmodule.{u2, u2} R R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (Semiring.toModule.{u2} R _inst_1)))
+Case conversion may be inaccurate. Consider using '#align ideal.torsion_of_zero Ideal.torsionOf_zeroₓ'. -/
 @[simp]
 theorem torsionOf_zero : torsionOf R M (0 : M) = ⊤ := by simp [torsion_of]
 #align ideal.torsion_of_zero Ideal.torsionOf_zero
 
 variable {R M}
 
+/- warning: ideal.mem_torsion_of_iff -> Ideal.mem_torsionOf_iff 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 ideal.mem_torsion_of_iff Ideal.mem_torsionOf_iffₓ'. -/
 @[simp]
 theorem mem_torsionOf_iff (x : M) (a : R) : a ∈ torsionOf R M x ↔ a • x = 0 :=
   Iff.rfl
@@ -90,6 +104,12 @@ theorem mem_torsionOf_iff (x : M) (a : R) : a ∈ torsionOf R M x ↔ a • x =
 
 variable (R)
 
+/- warning: ideal.torsion_of_eq_top_iff -> Ideal.torsionOf_eq_top_iff is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall (R : Type.{u2}) {M : Type.{u1}} [_inst_1 : Semiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M _inst_1 _inst_2] (m : M), Iff (Eq.{succ u2} (Ideal.{u2} R _inst_1) (Ideal.torsionOf.{u2, u1} R M _inst_1 _inst_2 _inst_3 m) (Top.top.{u2} (Ideal.{u2} R _inst_1) (Submodule.instTopSubmodule.{u2, u2} R R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (Semiring.toModule.{u2} R _inst_1)))) (Eq.{succ u1} M m (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2)))))
+Case conversion may be inaccurate. Consider using '#align ideal.torsion_of_eq_top_iff Ideal.torsionOf_eq_top_iffₓ'. -/
 @[simp]
 theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 :=
   by
@@ -98,6 +118,12 @@ theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 :=
   exact Submodule.mem_top
 #align ideal.torsion_of_eq_top_iff Ideal.torsionOf_eq_top_iff
 
+/- warning: ideal.torsion_of_eq_bot_iff_of_no_zero_smul_divisors -> Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisors is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M _inst_1 _inst_2] [_inst_4 : Nontrivial.{u1} R] [_inst_5 : NoZeroSMulDivisors.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))] (m : M), Iff (Eq.{succ u1} (Ideal.{u1} R _inst_1) (Ideal.torsionOf.{u1, u2} R M _inst_1 _inst_2 _inst_3 m) (Bot.bot.{u1} (Ideal.{u1} R _inst_1) (Submodule.hasBot.{u1, u1} R R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Semiring.toModule.{u1} R _inst_1)))) (Ne.{succ u2} M m (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))))))
+but is expected to have type
+  forall (R : Type.{u2}) {M : Type.{u1}} [_inst_1 : Semiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M _inst_1 _inst_2] [_inst_4 : Nontrivial.{u2} R] [_inst_5 : NoZeroSMulDivisors.{u2, u1} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2)) (SMulZeroClass.toSMul.{u2, u1} R M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R _inst_1) (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2)) (Module.toMulActionWithZero.{u2, u1} R M _inst_1 _inst_2 _inst_3))))] (m : M), Iff (Eq.{succ u2} (Ideal.{u2} R _inst_1) (Ideal.torsionOf.{u2, u1} R M _inst_1 _inst_2 _inst_3 m) (Bot.bot.{u2} (Ideal.{u2} R _inst_1) (Submodule.instBotSubmodule.{u2, u2} R R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (Semiring.toModule.{u2} R _inst_1)))) (Ne.{succ u1} M m (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2)))))
+Case conversion may be inaccurate. Consider using '#align ideal.torsion_of_eq_bot_iff_of_no_zero_smul_divisors Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisorsₓ'. -/
 @[simp]
 theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDivisors R M] (m : M) :
     torsionOf R M m = ⊥ ↔ m ≠ 0 :=
@@ -109,6 +135,12 @@ theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDiv
     tauto
 #align ideal.torsion_of_eq_bot_iff_of_no_zero_smul_divisors Ideal.torsionOf_eq_bot_iff_of_noZeroSMulDivisors
 
+/- warning: ideal.complete_lattice.independent.linear_independent' -> Ideal.CompleteLattice.Independent.linear_independent' 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}} {M : Type.{u1}} {v : ι -> M} [_inst_4 : Ring.{u2} R] [_inst_5 : AddCommGroup.{u1} M] [_inst_6 : Module.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M _inst_5)], (CompleteLattice.Independent.{succ u3, u1} ι (Submodule.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) _inst_6) (Submodule.completeLattice.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) _inst_6) (fun (i : ι) => Submodule.span.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) _inst_6 (Singleton.singleton.{u1, u1} M (Set.{u1} M) (Set.instSingletonSet.{u1} M) (v i)))) -> (forall (i : ι), Eq.{succ u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R _inst_4)) (Ideal.torsionOf.{u2, u1} R M (Ring.toSemiring.{u2} R _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) _inst_6 (v i)) (Bot.bot.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R _inst_4)) (Submodule.instBotSubmodule.{u2, u2} R R (Ring.toSemiring.{u2} R _inst_4) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_4)))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_4))))) -> (LinearIndependent.{u3, u2, u1} ι R M v (Ring.toSemiring.{u2} R _inst_4) (AddCommGroup.toAddCommMonoid.{u1} M _inst_5) _inst_6)
+Case conversion may be inaccurate. Consider using '#align ideal.complete_lattice.independent.linear_independent' Ideal.CompleteLattice.Independent.linear_independent'ₓ'. -/
 /-- See also `complete_lattice.independent.linear_independent` which provides the same conclusion
 but requires the stronger hypothesis `no_zero_smul_divisors R M`. -/
 theorem CompleteLattice.Independent.linear_independent' {ι R M : Type _} {v : ι → M} [Ring R]
@@ -134,14 +166,22 @@ section
 
 variable (R M : Type _) [Ring R] [AddCommGroup M] [Module R M]
 
+#print Ideal.quotTorsionOfEquivSpanSingleton /-
 /-- The span of `x` in `M` is isomorphic to `R` quotiented by the torsion ideal of `x`.-/
 noncomputable def quotTorsionOfEquivSpanSingleton (x : M) : (R ⧸ torsionOf R M x) ≃ₗ[R] R ∙ x :=
   (LinearMap.toSpanSingleton R M x).quotKerEquivRange.trans <|
     LinearEquiv.ofEq _ _ (LinearMap.span_singleton_eq_range R M x).symm
 #align ideal.quot_torsion_of_equiv_span_singleton Ideal.quotTorsionOfEquivSpanSingleton
+-/
 
 variable {R M}
 
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_inst_3 x) (Submodule.Quotient.mk.{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)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x) a)) (SMul.smul.{u1, u2} 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)) (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.hasSingleton.{u2} M) x))) (Submodule.smul.{u1, u1, u2} R R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.hasSingleton.{u2} M) x)) (Mul.toSMul.{u1} R (MulOneClass.toHasMul.{u1} R (Monoid.toMulOneClass.{u1} R (Ring.toMonoid.{u1} R _inst_1)))) (MulAction.toHasSmul.{u1, u2} R M (Ring.toMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (IsScalarTower.left.{u1, u2} R M (Ring.toMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) a (Subtype.mk.{succ u2} M (fun (x_1 : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.hasMem.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.hasSingleton.{u2} M) x))) x (Submodule.mem_span_singleton_self.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)))
+but is expected to have type
+  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)] (x : M) (a : R), Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) => Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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 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(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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) _x) (SMulHomClass.toFunLike.{max u1 u2, u1, u1, u2} (LinearEquiv.{u1, u1, u1, u2} 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))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) 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(Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M 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(Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddCommMonoid.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.Quotient.addCommGroup.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x))))) (DistribSMul.toSMulZeroClass.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddMonoid.toAddZeroClass.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddCommMonoid.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.Quotient.addCommGroup.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x))))) (DistribMulAction.toDistribSMul.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.Quotient.addCommGroup.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)))) (Module.toDistribMulAction.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.Quotient.addCommGroup.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x))) (Submodule.Quotient.module.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)))))) (SMulZeroClass.toSMul.{u1, u2} R (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (AddMonoid.toZero.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (AddCommMonoid.toAddMonoid.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))))) (DistribSMul.toSMulZeroClass.{u1, u2} R (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (AddMonoid.toAddZeroClass.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (AddCommMonoid.toAddMonoid.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))))) (DistribMulAction.toDistribSMul.{u1, u2} R (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (Module.toDistribMulAction.{u1, u2} R (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (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 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))))))) (DistribMulActionHomClass.toSMulHomClass.{max u1 u2, u1, u1, u2} (LinearEquiv.{u1, u1, u1, u2} 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))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (RingHomInvPair.ids.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.Quotient.addCommGroup.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x))) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))) (Submodule.Quotient.module.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Submodule.Quotient.addCommGroup.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)))) (AddCommMonoid.toAddMonoid.{u2} (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) 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_inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x))) (Submodule.Quotient.module.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x))) (Module.toDistribMulAction.{u1, u2} R (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (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 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (SemilinearMapClass.distribMulActionHomClass.{u1, u1, u2, max u1 u2} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Submodule.hasQuotient.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Ideal.torsionOf.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)) (Subtype.{succ u2} M (fun (x_1 : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (SetLike.instMembership.{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)) x_1 (Submodule.span.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x)))) (LinearEquiv.{u1, u1, u1, u2} 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))) (RingHom.id.{u1} R 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(Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{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_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{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_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) (IsScalarTower.left.{u1, u2} R M (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (MulActionWithZero.toMulAction.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R 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(Set.instSingletonSet.{u2} M) x))) x (Submodule.mem_span_singleton_self.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 x)))
+Case conversion may be inaccurate. Consider using '#align ideal.quot_torsion_of_equiv_span_singleton_apply_mk Ideal.quotTorsionOfEquivSpanSingleton_apply_mkₓ'. -/
 @[simp]
 theorem quotTorsionOfEquivSpanSingleton_apply_mk (x : M) (a : R) :
     quotTorsionOfEquivSpanSingleton R M x (Submodule.Quotient.mk a) =
@@ -161,19 +201,29 @@ variable (R M : Type _) [CommSemiring R] [AddCommMonoid M] [Module R M]
 
 namespace Submodule
 
+#print Submodule.torsionBy /-
 /-- The `a`-torsion submodule for `a` in `R`, containing all elements `x` of `M` such that
   `a • x = 0`. -/
 @[simps]
 def torsionBy (a : R) : Submodule R M :=
   (DistribMulAction.toLinearMap R M a).ker
 #align submodule.torsion_by Submodule.torsionBy
+-/
 
+#print Submodule.torsionBySet /-
 /-- The submodule containing all elements `x` of `M` such that `a • x = 0` for all `a` in `s`. -/
 @[simps]
 def torsionBySet (s : Set R) : Submodule R M :=
   sInf (torsionBy R M '' s)
 #align submodule.torsion_by_set Submodule.torsionBySet
+-/
 
+/- warning: submodule.torsion' -> Submodule.torsion' is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) (M : Type.{u2}) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (S : Type.{u3}) [_inst_4 : CommMonoid.{u3} S] [_inst_5 : DistribMulAction.{u3, u2} S M (CommMonoid.toMonoid.{u3} S _inst_4) (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] [_inst_6 : SMulCommClass.{u3, u1, u2} S R M (SMulZeroClass.toHasSmul.{u3, u2} S M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u3, u2} S M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u3, u2} S M (CommMonoid.toMonoid.{u3} S _inst_4) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_5))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} 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.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))))], Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3
+but is expected to have type
+  forall (R : Type.{u2}) (M : Type.{u3}) [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] (S : Type.{u1}) [_inst_4 : CommMonoid.{u1} S] [_inst_5 : DistribMulAction.{u1, u3} S M (CommMonoid.toMonoid.{u1} S _inst_4) (AddCommMonoid.toAddMonoid.{u3} M _inst_2)] [_inst_6 : SMulCommClass.{u1, u2, u3} S R M (SMulZeroClass.toSMul.{u1, u3} S M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u3} S M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u3} S M (CommMonoid.toMonoid.{u1} S _inst_4) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_5))) (SMulZeroClass.toSMul.{u2, u3} R M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R _inst_1)) (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Module.toMulActionWithZero.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3))))], Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3
+Case conversion may be inaccurate. Consider using '#align submodule.torsion' Submodule.torsion'ₓ'. -/
 /-- The `S`-torsion submodule, containing all elements `x` of `M` such that `a • x = 0` for some
 `a` in `S`. -/
 @[simps]
@@ -186,41 +236,51 @@ def torsion' (S : Type _) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S
   smul_mem' := fun a x ⟨b, h⟩ => ⟨b, by rw [smul_comm, h, smul_zero]⟩
 #align submodule.torsion' Submodule.torsion'
 
+#print Submodule.torsion /-
 /-- The torsion submodule, containing all elements `x` of `M` such that  `a • x = 0` for some
   non-zero-divisor `a` in `R`. -/
 @[reducible]
 def torsion :=
   torsion' R M R⁰
 #align submodule.torsion Submodule.torsion
+-/
 
 end Submodule
 
 namespace Module
 
+#print Module.IsTorsionBy /-
 /-- A `a`-torsion module is a module where every element is `a`-torsion. -/
 @[reducible]
 def IsTorsionBy (a : R) :=
   ∀ ⦃x : M⦄, a • x = 0
 #align module.is_torsion_by Module.IsTorsionBy
+-/
 
+#print Module.IsTorsionBySet /-
 /-- A module where every element is `a`-torsion for all `a` in `s`. -/
 @[reducible]
 def IsTorsionBySet (s : Set R) :=
   ∀ ⦃x : M⦄ ⦃a : s⦄, (a : R) • x = 0
 #align module.is_torsion_by_set Module.IsTorsionBySet
+-/
 
+#print Module.IsTorsion' /-
 /-- A `S`-torsion module is a module where every element is `a`-torsion for some `a` in `S`. -/
 @[reducible]
 def IsTorsion' (S : Type _) [SMul S M] :=
   ∀ ⦃x : M⦄, ∃ a : S, a • x = 0
 #align module.is_torsion' Module.IsTorsion'
+-/
 
+#print Module.IsTorsion /-
 /-- A torsion module is a module where every element is `a`-torsion for some non-zero-divisor `a`.
 -/
 @[reducible]
 def IsTorsion :=
   ∀ ⦃x : M⦄, ∃ a : R⁰, a • x = 0
 #align module.is_torsion Module.IsTorsion
+-/
 
 end Module
 
@@ -234,21 +294,41 @@ variable [CommSemiring R] [AddCommMonoid M] [Module R M] (s : Set R) (a : R)
 
 namespace Submodule
 
+#print Submodule.smul_torsionBy /-
 @[simp]
 theorem smul_torsionBy (x : torsionBy R M a) : a • x = 0 :=
   Subtype.ext x.Prop
 #align submodule.smul_torsion_by Submodule.smul_torsionBy
+-/
 
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+Case conversion may be inaccurate. Consider using '#align submodule.smul_coe_torsion_by Submodule.smul_coe_torsionByₓ'. -/
 @[simp]
 theorem smul_coe_torsionBy (x : torsionBy R M a) : a • (x : M) = 0 :=
   x.Prop
 #align submodule.smul_coe_torsion_by Submodule.smul_coe_torsionBy
 
+/- warning: submodule.mem_torsion_by_iff -> Submodule.mem_torsionBy_iff is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align submodule.mem_torsion_by_iff Submodule.mem_torsionBy_iffₓ'. -/
 @[simp]
 theorem mem_torsionBy_iff (x : M) : x ∈ torsionBy R M a ↔ a • x = 0 :=
   Iff.rfl
 #align submodule.mem_torsion_by_iff Submodule.mem_torsionBy_iff
 
+/- warning: submodule.mem_torsion_by_set_iff -> Submodule.mem_torsionBySet_iff is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align submodule.mem_torsion_by_set_iff Submodule.mem_torsionBySet_iffₓ'. -/
 @[simp]
 theorem mem_torsionBySet_iff (x : M) : x ∈ torsionBySet R M s ↔ ∀ a : s, (a : R) • x = 0 :=
   by
@@ -256,21 +336,30 @@ theorem mem_torsionBySet_iff (x : M) : x ∈ torsionBySet R M s ↔ ∀ a : s, (
   rintro _ ⟨a, ha, rfl⟩; exact h ⟨a, ha⟩
 #align submodule.mem_torsion_by_set_iff Submodule.mem_torsionBySet_iff
 
+#print Submodule.torsionBySet_singleton_eq /-
 @[simp]
-theorem torsionBy_singleton_eq : torsionBySet R M {a} = torsionBy R M a :=
+theorem torsionBySet_singleton_eq : torsionBySet R M {a} = torsionBy R M a :=
   by
   ext x
   simp only [mem_torsion_by_set_iff, SetCoe.forall, Subtype.coe_mk, Set.mem_singleton_iff,
     forall_eq, mem_torsion_by_iff]
-#align submodule.torsion_by_singleton_eq Submodule.torsionBy_singleton_eq
+#align submodule.torsion_by_singleton_eq Submodule.torsionBySet_singleton_eq
+-/
 
+/- warning: submodule.torsion_by_set_le_torsion_by_set_of_subset -> Submodule.torsionBySet_le_torsionBySet_of_subset is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] {s : Set.{u2} R} {t : Set.{u2} R}, (HasSubset.Subset.{u2} (Set.{u2} R) (Set.instHasSubsetSet.{u2} R) s t) -> (LE.le.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3))))) (Submodule.torsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 t) (Submodule.torsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 s))
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_set_le_torsion_by_set_of_subset Submodule.torsionBySet_le_torsionBySet_of_subsetₓ'. -/
 theorem torsionBySet_le_torsionBySet_of_subset {s t : Set R} (st : s ⊆ t) :
     torsionBySet R M t ≤ torsionBySet R M s :=
   sInf_le_sInf fun _ ⟨a, ha, h⟩ => ⟨a, st ha, h⟩
 #align submodule.torsion_by_set_le_torsion_by_set_of_subset Submodule.torsionBySet_le_torsionBySet_of_subset
 
+#print Submodule.torsionBySet_eq_torsionBySet_span /-
 /-- Torsion by a set is torsion by the ideal generated by it. -/
-theorem torsionBySet_eq_torsion_by_span : torsionBySet R M s = torsionBySet R M (Ideal.span s) :=
+theorem torsionBySet_eq_torsionBySet_span : torsionBySet R M s = torsionBySet R M (Ideal.span s) :=
   by
   refine' le_antisymm (fun x hx => _) (torsion_by_set_le_torsion_by_set_of_subset subset_span)
   rw [mem_torsion_by_set_iff] at hx⊢
@@ -280,12 +369,21 @@ theorem torsionBySet_eq_torsion_by_span : torsionBySet R M s = torsionBySet R M
     exact this ha
   rw [Ideal.span_le]
   exact fun a ha => hx ⟨a, ha⟩
-#align submodule.torsion_by_set_eq_torsion_by_span Submodule.torsionBySet_eq_torsion_by_span
+#align submodule.torsion_by_set_eq_torsion_by_span Submodule.torsionBySet_eq_torsionBySet_span
+-/
 
-theorem torsionBy_span_singleton_eq : torsionBySet R M (R ∙ a) = torsionBy R M a :=
-  (torsionBySet_eq_torsion_by_span _).symm.trans <| torsionBy_singleton_eq _
-#align submodule.torsion_by_span_singleton_eq Submodule.torsionBy_span_singleton_eq
+#print Submodule.torsionBySet_span_singleton_eq /-
+theorem torsionBySet_span_singleton_eq : torsionBySet R M (R ∙ a) = torsionBy R M a :=
+  (torsionBySet_eq_torsionBySet_span _).symm.trans <| torsionBySet_singleton_eq _
+#align submodule.torsion_by_span_singleton_eq Submodule.torsionBySet_span_singleton_eq
+-/
 
+/- warning: submodule.torsion_by_le_torsion_by_of_dvd -> Submodule.torsionBy_le_torsionBy_of_dvd is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (a : R) (b : R), (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (CommSemiring.toNonUnitalCommSemiring.{u1} R _inst_1))))) a b) -> (LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))))) (Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 a) (Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 b))
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] (a : R) (b : R), (Dvd.dvd.{u2} R (semigroupDvd.{u2} R (SemigroupWithZero.toSemigroup.{u2} R (NonUnitalSemiring.toSemigroupWithZero.{u2} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u2} R (CommSemiring.toNonUnitalCommSemiring.{u2} R _inst_1))))) a b) -> (LE.le.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Preorder.toLE.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3))))) (Submodule.torsionBy.{u2, u1} R M _inst_1 _inst_2 _inst_3 a) (Submodule.torsionBy.{u2, u1} R M _inst_1 _inst_2 _inst_3 b))
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_le_torsion_by_of_dvd Submodule.torsionBy_le_torsionBy_of_dvdₓ'. -/
 theorem torsionBy_le_torsionBy_of_dvd (a b : R) (dvd : a ∣ b) : torsionBy R M a ≤ torsionBy R M b :=
   by
   rw [← torsion_by_span_singleton_eq, ← torsion_by_singleton_eq]
@@ -293,6 +391,12 @@ theorem torsionBy_le_torsionBy_of_dvd (a b : R) (dvd : a ∣ b) : torsionBy R M
   rintro c (rfl : c = b); exact ideal.mem_span_singleton.mpr dvd
 #align submodule.torsion_by_le_torsion_by_of_dvd Submodule.torsionBy_le_torsionBy_of_dvd
 
+/- warning: submodule.torsion_by_one -> Submodule.torsionBy_one is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2], Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))))) (Bot.bot.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.hasBot.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2], Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Bot.bot.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.instBotSubmodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_one Submodule.torsionBy_oneₓ'. -/
 @[simp]
 theorem torsionBy_one : torsionBy R M 1 = ⊥ :=
   eq_bot_iff.mpr fun _ h => by
@@ -300,12 +404,18 @@ theorem torsionBy_one : torsionBy R M 1 = ⊥ :=
     exact h
 #align submodule.torsion_by_one Submodule.torsionBy_one
 
+/- warning: submodule.torsion_by_univ -> Submodule.torsionBySet_univ is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2], Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Set.univ.{u1} R)) (Bot.bot.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.hasBot.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2], Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Set.univ.{u1} R)) (Bot.bot.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.instBotSubmodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_univ Submodule.torsionBySet_univₓ'. -/
 @[simp]
-theorem torsion_by_univ : torsionBySet R M Set.univ = ⊥ :=
+theorem torsionBySet_univ : torsionBySet R M Set.univ = ⊥ :=
   by
   rw [eq_bot_iff, ← torsion_by_one, ← torsion_by_singleton_eq]
   exact torsion_by_set_le_torsion_by_set_of_subset fun _ _ => trivial
-#align submodule.torsion_by_univ Submodule.torsion_by_univ
+#align submodule.torsion_by_univ Submodule.torsionBySet_univ
 
 end Submodule
 
@@ -313,13 +423,25 @@ open Submodule
 
 namespace Module
 
+/- warning: module.is_torsion_by_singleton_iff -> Module.isTorsionBySet_singleton_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (a : R), Iff (Module.IsTorsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) a)) (Module.IsTorsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 a)
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] (a : R), Iff (Module.IsTorsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 (Singleton.singleton.{u2, u2} R (Set.{u2} R) (Set.instSingletonSet.{u2} R) a)) (Module.IsTorsionBy.{u2, u1} R M _inst_1 _inst_2 _inst_3 a)
+Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_singleton_iff Module.isTorsionBySet_singleton_iffₓ'. -/
 @[simp]
-theorem isTorsionBy_singleton_iff : IsTorsionBySet R M {a} ↔ IsTorsionBy R M a :=
+theorem isTorsionBySet_singleton_iff : IsTorsionBySet R M {a} ↔ IsTorsionBy R M a :=
   by
   refine' ⟨fun h x => @h _ ⟨_, Set.mem_singleton _⟩, fun h x => _⟩
   rintro ⟨b, rfl : b = a⟩; exact @h _
-#align module.is_torsion_by_singleton_iff Module.isTorsionBy_singleton_iff
-
+#align module.is_torsion_by_singleton_iff Module.isTorsionBySet_singleton_iff
+
+/- warning: module.is_torsion_by_set_iff_torsion_by_set_eq_top -> Module.isTorsionBySet_iff_torsionBySet_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (s : Set.{u1} R), Iff (Module.IsTorsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s) (Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s) (Top.top.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.hasTop.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)))
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] (s : Set.{u2} R), Iff (Module.IsTorsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 s) (Eq.{succ u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Submodule.torsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 s) (Top.top.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Submodule.instTopSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3)))
+Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_set_iff_torsion_by_set_eq_top Module.isTorsionBySet_iff_torsionBySet_eq_topₓ'. -/
 theorem isTorsionBySet_iff_torsionBySet_eq_top :
     IsTorsionBySet R M s ↔ Submodule.torsionBySet R M s = ⊤ :=
   ⟨fun h => eq_top_iff.mpr fun _ _ => (mem_torsionBySet_iff _ _).mpr <| @h _, fun h x =>
@@ -328,21 +450,39 @@ theorem isTorsionBySet_iff_torsionBySet_eq_top :
     trivial⟩
 #align module.is_torsion_by_set_iff_torsion_by_set_eq_top Module.isTorsionBySet_iff_torsionBySet_eq_top
 
+/- warning: module.is_torsion_by_iff_torsion_by_eq_top -> Module.isTorsionBy_iff_torsionBy_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (a : R), Iff (Module.IsTorsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 a) (Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 a) (Top.top.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.hasTop.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)))
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] (a : R), Iff (Module.IsTorsionBy.{u2, u1} R M _inst_1 _inst_2 _inst_3 a) (Eq.{succ u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Submodule.torsionBy.{u2, u1} R M _inst_1 _inst_2 _inst_3 a) (Top.top.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Submodule.instTopSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3)))
+Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_iff_torsion_by_eq_top Module.isTorsionBy_iff_torsionBy_eq_topₓ'. -/
 /-- A `a`-torsion module is a module whose `a`-torsion submodule is the full space. -/
 theorem isTorsionBy_iff_torsionBy_eq_top : IsTorsionBy R M a ↔ torsionBy R M a = ⊤ := by
   rw [← torsion_by_singleton_eq, ← is_torsion_by_singleton_iff,
     is_torsion_by_set_iff_torsion_by_set_eq_top]
 #align module.is_torsion_by_iff_torsion_by_eq_top Module.isTorsionBy_iff_torsionBy_eq_top
 
+/- warning: module.is_torsion_by_set_iff_is_torsion_by_span -> Module.isTorsionBySet_iff_is_torsion_by_span is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (s : Set.{u1} R), Iff (Module.IsTorsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s) (Module.IsTorsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (Ideal.span.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1) s)))
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] (s : Set.{u2} R), Iff (Module.IsTorsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 s) (Module.IsTorsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 (SetLike.coe.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) (Ideal.span.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1) s)))
+Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_set_iff_is_torsion_by_span Module.isTorsionBySet_iff_is_torsion_by_spanₓ'. -/
 theorem isTorsionBySet_iff_is_torsion_by_span :
     IsTorsionBySet R M s ↔ IsTorsionBySet R M (Ideal.span s) := by
   rw [is_torsion_by_set_iff_torsion_by_set_eq_top, is_torsion_by_set_iff_torsion_by_set_eq_top,
     torsion_by_set_eq_torsion_by_span]
 #align module.is_torsion_by_set_iff_is_torsion_by_span Module.isTorsionBySet_iff_is_torsion_by_span
 
-theorem isTorsionBy_span_singleton_iff : IsTorsionBySet R M (R ∙ a) ↔ IsTorsionBy R M a :=
-  (isTorsionBySet_iff_is_torsion_by_span _).symm.trans <| isTorsionBy_singleton_iff _
-#align module.is_torsion_by_span_singleton_iff Module.isTorsionBy_span_singleton_iff
+/- warning: module.is_torsion_by_span_singleton_iff -> Module.isTorsionBySet_span_singleton_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (a : R), Iff (Module.IsTorsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Submodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Submodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Submodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Submodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) a)))) (Module.IsTorsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 a)
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] (a : R), Iff (Module.IsTorsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 (SetLike.coe.{u2, u2} (Submodule.{u2, u2} R R (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) (Submodule.span.{u2, u2} R R (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (Singleton.singleton.{u2, u2} R (Set.{u2} R) (Set.instSingletonSet.{u2} R) a)))) (Module.IsTorsionBy.{u2, u1} R M _inst_1 _inst_2 _inst_3 a)
+Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_span_singleton_iff Module.isTorsionBySet_span_singleton_iffₓ'. -/
+theorem isTorsionBySet_span_singleton_iff : IsTorsionBySet R M (R ∙ a) ↔ IsTorsionBy R M a :=
+  (isTorsionBySet_iff_is_torsion_by_span _).symm.trans <| isTorsionBySet_singleton_iff _
+#align module.is_torsion_by_span_singleton_iff Module.isTorsionBySet_span_singleton_iff
 
 end Module
 
@@ -350,19 +490,43 @@ namespace Submodule
 
 open Module
 
+/- warning: submodule.torsion_by_set_is_torsion_by_set -> Submodule.torsionBySet_isTorsionBySet is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (s : Set.{u1} R), Module.IsTorsionBySet.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s)) _inst_1 (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s)) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s)) s
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] (s : Set.{u2} R), Module.IsTorsionBySet.{u2, u1} R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 s))) _inst_1 (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 s)) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 s)) s
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_set_is_torsion_by_set Submodule.torsionBySet_isTorsionBySetₓ'. -/
 theorem torsionBySet_isTorsionBySet : IsTorsionBySet R (torsionBySet R M s) s := fun ⟨x, hx⟩ a =>
   Subtype.ext <| (mem_torsionBySet_iff _ _).mp hx a
 #align submodule.torsion_by_set_is_torsion_by_set Submodule.torsionBySet_isTorsionBySet
 
+/- warning: submodule.torsion_by_is_torsion_by -> Submodule.torsionBy_isTorsionBy is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (a : R), Module.IsTorsionBy.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 a)) _inst_1 (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 a)) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 a)) a
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] (a : R), Module.IsTorsionBy.{u2, u1} R (Subtype.{succ u1} M (fun (x : M) => Membership.mem.{u1, u1} M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsionBy.{u2, u1} R M _inst_1 _inst_2 _inst_3 a))) _inst_1 (Submodule.addCommMonoid.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBy.{u2, u1} R M _inst_1 _inst_2 _inst_3 a)) (Submodule.module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBy.{u2, u1} R M _inst_1 _inst_2 _inst_3 a)) a
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_is_torsion_by Submodule.torsionBy_isTorsionByₓ'. -/
 /-- The `a`-torsion submodule is a `a`-torsion module. -/
 theorem torsionBy_isTorsionBy : IsTorsionBy R (torsionBy R M a) a := fun _ => smul_torsionBy _ _
 #align submodule.torsion_by_is_torsion_by Submodule.torsionBy_isTorsionBy
 
+/- warning: submodule.torsion_by_torsion_by_eq_top -> Submodule.torsionBy_torsionBy_eq_top is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_torsion_by_eq_top Submodule.torsionBy_torsionBy_eq_topₓ'. -/
 @[simp]
 theorem torsionBy_torsionBy_eq_top : torsionBy R (torsionBy R M a) a = ⊤ :=
   (isTorsionBy_iff_torsionBy_eq_top a).mp <| torsionBy_isTorsionBy a
 #align submodule.torsion_by_torsion_by_eq_top Submodule.torsionBy_torsionBy_eq_top
 
+/- warning: submodule.torsion_by_set_torsion_by_set_eq_top -> Submodule.torsionBySet_torsionBySet_eq_top is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (s : Set.{u1} R), Eq.{succ u2} (Submodule.{u1, u2} R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s))) (CommSemiring.toSemiring.{u1} R _inst_1) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s)) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s))) (Submodule.torsionBySet.{u1, u2} R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s))) _inst_1 (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s)) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s)) s) (Top.top.{u2} (Submodule.{u1, u2} R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s))) (CommSemiring.toSemiring.{u1} R _inst_1) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s)) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s))) (Submodule.instTopSubmodule.{u1, u2} R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s))) (CommSemiring.toSemiring.{u1} R _inst_1) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s)) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 s))))
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_set_torsion_by_set_eq_top Submodule.torsionBySet_torsionBySet_eq_topₓ'. -/
 @[simp]
 theorem torsionBySet_torsionBySet_eq_top : torsionBySet R (torsionBySet R M s) s = ⊤ :=
   (isTorsionBySet_iff_torsionBySet_eq_top s).mp <| torsionBySet_isTorsionBySet s
@@ -370,6 +534,7 @@ theorem torsionBySet_torsionBySet_eq_top : torsionBySet R (torsionBySet R M s) s
 
 variable (R M)
 
+#print Submodule.torsion_gc /-
 theorem torsion_gc :
     @GaloisConnection (Submodule R M) (Ideal R)ᵒᵈ _ _ annihilator fun I =>
       torsionBySet R M <| I.ofDual :=
@@ -377,6 +542,7 @@ theorem torsion_gc :
   ⟨fun h x hx => (mem_torsionBySet_iff _ _).mpr fun ⟨a, ha⟩ => mem_annihilator.mp (h ha) x hx,
     fun h a ha => mem_annihilator.mpr fun x hx => (mem_torsionBySet_iff _ _).mp (h hx) ⟨a, ha⟩⟩
 #align submodule.torsion_gc Submodule.torsion_gc
+-/
 
 variable {R M}
 
@@ -390,7 +556,13 @@ variable (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
 
 include hp
 
-theorem iSup_torsion_by_ideal_eq_torsion_by_iInf :
+/- warning: submodule.supr_torsion_by_ideal_eq_torsion_by_infi -> Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf is a dubious translation:
+lean 3 declaration is
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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i S) (fun (H : Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i S) => p i))))))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] {ι : Type.{u3}} {p : ι -> (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))} {S : Finset.{u3} ι}, (Set.Pairwise.{u3} ι (Finset.toSet.{u3} ι S) (fun (i : ι) (j : ι) => Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Sup.sup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SemilatticeSup.toSup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (IdemCommSemiring.toSemilatticeSup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Ideal.instIdemCommSemiringIdealToSemiring.{u1} R _inst_1))) (p i) (p j)) (Top.top.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.instTopSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) -> (forall [inst._@.Mathlib.Algebra.Module.Torsion._hyg.2869 : DecidableEq.{succ u3} ι], Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (iSup.{u2, succ u3} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (ConditionallyCompleteLattice.toSupSet.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) ι (fun (i : ι) => iSup.{u2, 0} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (ConditionallyCompleteLattice.toSupSet.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i S) (fun (h._@.Mathlib.Algebra.Module.Torsion._hyg.2878 : Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i S) => Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (p i))))) (Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (iInf.{u1, succ u3} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.instInfSetSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) ι (fun (i : ι) => iInf.{u1, 0} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.instInfSetSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i S) (fun (h._@.Mathlib.Algebra.Module.Torsion._hyg.2919 : Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i S) => p i))))))
+Case conversion may be inaccurate. Consider using '#align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInfₓ'. -/
+theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
     (⨆ i ∈ S, torsionBySet R M <| p i) = torsionBySet R M ↑(⨅ i ∈ S, p i) :=
   by
   cases' S.eq_empty_or_nonempty with h h
@@ -429,9 +601,15 @@ theorem iSup_torsion_by_ideal_eq_torsion_by_iInf :
         exact Ideal.mul_mem_left _ _ (this j hj ij)
     · simp_rw [coe_mk]
       rw [← Finset.sum_smul, hμ, one_smul]
-#align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsion_by_ideal_eq_torsion_by_iInf
-
-theorem supIndep_torsion_by_ideal : S.SupIndep fun i => torsionBySet R M <| p i :=
+#align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf
+
+/- warning: submodule.sup_indep_torsion_by_ideal -> Submodule.supIndep_torsionBySet_ideal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] {ι : Type.{u3}} {p : ι -> (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))} {S : Finset.{u3} ι}, (Set.Pairwise.{u3} ι ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} ι) (Set.{u3} ι) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} ι) (Set.{u3} ι) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} ι) (Set.{u3} ι) (Finset.Set.hasCoeT.{u3} ι))) S) (fun (i : ι) (j : ι) => Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Sup.sup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SemilatticeSup.toHasSup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (IdemSemiring.toSemilatticeSup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.idemSemiring.{u1, u1} R _inst_1 R (CommSemiring.toSemiring.{u1} R _inst_1) (Algebra.id.{u1} R _inst_1)))) (p i) (p j)) (Top.top.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.hasTop.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) -> (Finset.SupIndep.{u2, u3} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) ι (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) (Submodule.orderBot.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) S (fun (i : ι) => Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (p i))))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] {ι : Type.{u3}} {p : ι -> (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))} {S : Finset.{u3} ι}, (Set.Pairwise.{u3} ι (Finset.toSet.{u3} ι S) (fun (i : ι) (j : ι) => Eq.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Sup.sup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (SemilatticeSup.toSup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (IdemCommSemiring.toSemilatticeSup.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Ideal.instIdemCommSemiringIdealToSemiring.{u1} R _inst_1))) (p i) (p j)) (Top.top.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.instTopSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) -> (forall [inst._@.Mathlib.Algebra.Module.Torsion._hyg.3647 : DecidableEq.{succ u3} ι], Finset.SupIndep.{u2, u3} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) ι (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) (Submodule.instOrderBotSubmoduleToLEToPreorderInstPartialOrderSetLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) S (fun (i : ι) => Submodule.torsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (p i))))
+Case conversion may be inaccurate. Consider using '#align submodule.sup_indep_torsion_by_ideal Submodule.supIndep_torsionBySet_idealₓ'. -/
+theorem supIndep_torsionBySet_ideal : S.SupIndep fun i => torsionBySet R M <| p i :=
   fun T hT i hi hiT =>
   by
   rw [disjoint_iff, Finset.sup_eq_iSup,
@@ -441,7 +619,7 @@ theorem supIndep_torsion_by_ideal : S.SupIndep fun i => torsionBySet R M <| p i
   dsimp at this⊢
   rw [← this, Ideal.sup_iInf_eq_top, top_coe, torsion_by_univ]
   intro j hj; apply hp hi (hT hj); rintro rfl; exact hiT hj
-#align submodule.sup_indep_torsion_by_ideal Submodule.supIndep_torsion_by_ideal
+#align submodule.sup_indep_torsion_by_ideal Submodule.supIndep_torsionBySet_ideal
 
 omit hp
 
@@ -449,6 +627,12 @@ variable {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCoprime on q))
 
 include hq
 
+/- warning: submodule.supr_torsion_by_eq_torsion_by_prod -> Submodule.iSup_torsionBy_eq_torsionBy_prod is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] {ι : Type.{u3}} {S : Finset.{u3} ι} {q : ι -> R}, (Set.Pairwise.{u3} ι ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} ι) (Set.{u3} ι) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} ι) (Set.{u3} ι) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} ι) (Set.{u3} ι) (Finset.Set.hasCoeT.{u3} ι))) S) (Function.onFun.{succ u3, succ u1, 1} ι R Prop (IsCoprime.{u1} R _inst_1) q)) -> (Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (iSup.{u2, succ u3} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (ConditionallyCompleteLattice.toHasSup.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) ι (fun (i : ι) => iSup.{u2, 0} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (ConditionallyCompleteLattice.toHasSup.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i S) (fun (H : Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i S) => Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 (q i)))) (Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Finset.prod.{u1, u3} R ι (CommSemiring.toCommMonoid.{u1} R _inst_1) S (fun (i : ι) => q i))))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] {ι : Type.{u3}} {S : Finset.{u3} ι} {q : ι -> R}, (Set.Pairwise.{u3} ι (Finset.toSet.{u3} ι S) (Function.onFun.{succ u3, succ u1, 1} ι R Prop (IsCoprime.{u1} R _inst_1) q)) -> (forall [inst._@.Mathlib.Algebra.Module.Torsion._hyg.3961 : DecidableEq.{succ u3} ι], Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (iSup.{u2, succ u3} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (ConditionallyCompleteLattice.toSupSet.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) ι (fun (i : ι) => iSup.{u2, 0} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (ConditionallyCompleteLattice.toSupSet.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) (Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i S) (fun (h._@.Mathlib.Algebra.Module.Torsion._hyg.3970 : Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) i S) => Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 (q i)))) (Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Finset.prod.{u1, u3} R ι (CommSemiring.toCommMonoid.{u1} R _inst_1) S (fun (i : ι) => q i))))
+Case conversion may be inaccurate. Consider using '#align submodule.supr_torsion_by_eq_torsion_by_prod Submodule.iSup_torsionBy_eq_torsionBy_prodₓ'. -/
 theorem iSup_torsionBy_eq_torsionBy_prod :
     (⨆ i ∈ S, torsionBy R M <| q i) = torsionBy R M (∏ i in S, q i) :=
   by
@@ -463,6 +647,12 @@ theorem iSup_torsionBy_eq_torsionBy_prod :
   · exact fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime _ _).mpr (hq hi hj ij)
 #align submodule.supr_torsion_by_eq_torsion_by_prod Submodule.iSup_torsionBy_eq_torsionBy_prod
 
+/- warning: submodule.sup_indep_torsion_by -> Submodule.supIndep_torsionBy is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] {ι : Type.{u3}} {S : Finset.{u3} ι} {q : ι -> R}, (Set.Pairwise.{u3} ι ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} ι) (Set.{u3} ι) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} ι) (Set.{u3} ι) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} ι) (Set.{u3} ι) (Finset.Set.hasCoeT.{u3} ι))) S) (Function.onFun.{succ u3, succ u1, 1} ι R Prop (IsCoprime.{u1} R _inst_1) q)) -> (Finset.SupIndep.{u2, u3} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) ι (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) (Submodule.orderBot.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) S (fun (i : ι) => Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 (q i)))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] {ι : Type.{u3}} {S : Finset.{u3} ι} {q : ι -> R}, (Set.Pairwise.{u3} ι (Finset.toSet.{u3} ι S) (Function.onFun.{succ u3, succ u1, 1} ι R Prop (IsCoprime.{u1} R _inst_1) q)) -> (forall [inst._@.Mathlib.Algebra.Module.Torsion._hyg.4151 : DecidableEq.{succ u3} ι], Finset.SupIndep.{u2, u3} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) ι (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) (Submodule.instOrderBotSubmoduleToLEToPreorderInstPartialOrderSetLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) S (fun (i : ι) => Submodule.torsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 (q i)))
+Case conversion may be inaccurate. Consider using '#align submodule.sup_indep_torsion_by Submodule.supIndep_torsionByₓ'. -/
 theorem supIndep_torsionBy : S.SupIndep fun i => torsionBy R M <| q i :=
   by
   convert sup_indep_torsion_by_ideal fun i hi j hj ij =>
@@ -486,6 +676,12 @@ open BigOperators
 
 variable {ι : Type _} [DecidableEq ι] {S : Finset ι}
 
+/- warning: submodule.torsion_by_set_is_internal -> Submodule.torsionBySet_isInternal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}} [_inst_4 : DecidableEq.{succ u3} ι] {S : Finset.{u3} ι} {p : ι -> (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))}, (Set.Pairwise.{u3} ι ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} ι) (Set.{u3} ι) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} ι) (Set.{u3} ι) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} ι) (Set.{u3} ι) (Finset.Set.hasCoeT.{u3} ι))) S) (fun (i : ι) (j : ι) => Eq.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Sup.sup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SemilatticeSup.toHasSup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (IdemSemiring.toSemilatticeSup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.idemSemiring.{u1, u1} R (CommRing.toCommSemiring.{u1} R _inst_1) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (p i) (p j)) (Top.top.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasTop.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) -> (Module.IsTorsionBySet.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{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)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (iInf.{u1, succ u3} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasInf.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) ι (fun (i : ι) => iInf.{u1, 0} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasInf.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i S) (fun (H : Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) i S) => p i))))) -> (DirectSum.IsInternal.{u3, u2, u2} (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (fun (a : coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) (b : coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) => Subtype.decidableEq.{u3} ι (fun (x : ι) => Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) x S) (fun (a : ι) (b : ι) => _inst_4 a b) a b) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.addSubmonoidClass.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (fun (i : coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) => Submodule.torsionBySet.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{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)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (p ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) ι (HasLiftT.mk.{succ u3, succ u3} (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) ι (CoeTCₓ.coe.{succ u3, succ u3} (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) ι (coeBase.{succ u3, succ u3} (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) ι (coeSubtype.{succ u3} ι (fun (x : ι) => Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) x S))))) i)))))
+but is expected to have type
+  forall {R : Type.{u3}} {M : Type.{u1}} [_inst_1 : CommRing.{u3} R] [_inst_2 : AddCommGroup.{u1} M] [_inst_3 : Module.{u3, u1} R M (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2)] {ι : Type.{u2}} [_inst_4 : DecidableEq.{succ u2} ι] {S : Finset.{u2} ι} {p : ι -> (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)))}, (Set.Pairwise.{u2} ι (Finset.toSet.{u2} ι S) (fun (i : ι) (j : ι) => Eq.{succ u3} (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Sup.sup.{u3} (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (SemilatticeSup.toSup.{u3} (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (IdemCommSemiring.toSemilatticeSup.{u3} (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Ideal.instIdemCommSemiringIdealToSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)))) (p i) (p j)) (Top.top.{u3} (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Submodule.instTopSubmodule.{u3, u3} R R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))))))) -> (Module.IsTorsionBySet.{u3, u1} R M (CommRing.toCommSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (SetLike.coe.{u3, u3} (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) R (Submodule.setLike.{u3, u3} R R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)))) (iInf.{u3, succ u2} (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Submodule.instInfSetSubmodule.{u3, u3} R R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)))) ι (fun (i : ι) => iInf.{u3, 0} (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Submodule.instInfSetSubmodule.{u3, u3} R R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)))) (Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) i S) (fun (H : Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) i S) => p i))))) -> (DirectSum.IsInternal.{u2, u1, u1} (Subtype.{succ u2} ι (fun (x : ι) => Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) x S)) M (Submodule.{u3, u1} R M (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) (fun (a : Subtype.{succ u2} ι (fun (x : ι) => Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) x S)) (b : Subtype.{succ u2} ι (fun (x : ι) => Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) x S)) => Subtype.instDecidableEqSubtype.{u2} ι (fun (x : ι) => Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) x S) (fun (a : ι) (b : ι) => _inst_4 a b) a b) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (Submodule.setLike.{u3, u1} R M (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) (Submodule.addSubmonoidClass.{u3, u1} R M (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) (fun (i : Subtype.{succ u2} ι (fun (x : ι) => Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) x S)) => Submodule.torsionBySet.{u3, u1} R M (CommRing.toCommSemiring.{u3} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (SetLike.coe.{u3, u3} (Ideal.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) R (Submodule.setLike.{u3, u3} R R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)))) (p (Subtype.val.{succ u2} ι (fun (x : ι) => Membership.mem.{u2, u2} ι (Finset.{u2} ι) (Finset.instMembershipFinset.{u2} ι) x S) i)))))
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_set_is_internal Submodule.torsionBySet_isInternalₓ'. -/
 /-- If the `p i` are pairwise coprime, a `⨅ i, p i`-torsion module is the internal direct sum of
 its `p i`-torsion submodules.-/
 theorem torsionBySet_isInternal {p : ι → Ideal R}
@@ -493,19 +689,25 @@ theorem torsionBySet_isInternal {p : ι → Ideal R}
     (hM : Module.IsTorsionBySet R M (⨅ i ∈ S, p i : Ideal R)) :
     DirectSum.IsInternal fun i : S => torsionBySet R M <| p i :=
   DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top
-    (CompleteLattice.independent_iff_supIndep.mpr <| supIndep_torsion_by_ideal hp)
+    (CompleteLattice.independent_iff_supIndep.mpr <| supIndep_torsionBySet_ideal hp)
     ((iSup_subtype'' ↑S fun i => torsionBySet R M <| p i).trans <|
-      (iSup_torsion_by_ideal_eq_torsion_by_iInf hp).trans <|
+      (iSup_torsionBySet_ideal_eq_torsionBySet_iInf hp).trans <|
         (Module.isTorsionBySet_iff_torsionBySet_eq_top _).mp hM)
 #align submodule.torsion_by_set_is_internal Submodule.torsionBySet_isInternal
 
+/- warning: submodule.torsion_by_is_internal -> Submodule.torsionBy_isInternal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {ι : Type.{u3}} [_inst_4 : DecidableEq.{succ u3} ι] {S : Finset.{u3} ι} {q : ι -> R}, (Set.Pairwise.{u3} ι ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} ι) (Set.{u3} ι) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} ι) (Set.{u3} ι) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} ι) (Set.{u3} ι) (Finset.Set.hasCoeT.{u3} ι))) S) (Function.onFun.{succ u3, succ u1, 1} ι R Prop (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) q)) -> (Module.IsTorsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Finset.prod.{u1, u3} R ι (CommRing.toCommMonoid.{u1} R _inst_1) S (fun (i : ι) => q i))) -> (DirectSum.IsInternal.{u3, u2, u2} (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (fun (a : coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) (b : coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) => Subtype.decidableEq.{u3} ι (fun (x : ι) => Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) x S) (fun (a : ι) (b : ι) => _inst_4 a b) a b) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.addSubmonoidClass.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (fun (i : coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) => Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (q ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) ι (HasLiftT.mk.{succ u3, succ u3} (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) ι (CoeTCₓ.coe.{succ u3, succ u3} (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) ι (coeBase.{succ u3, succ u3} (coeSort.{succ u3, succ (succ u3)} (Finset.{u3} ι) Type.{u3} (Finset.hasCoeToSort.{u3} ι) S) ι (coeSubtype.{succ u3} ι (fun (x : ι) => Membership.Mem.{u3, u3} ι (Finset.{u3} ι) (Finset.hasMem.{u3} ι) x S))))) i))))
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : AddCommGroup.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2)] {ι : Type.{u3}} [_inst_4 : DecidableEq.{succ u3} ι] {S : Finset.{u3} ι} {q : ι -> R}, (Set.Pairwise.{u3} ι (Finset.toSet.{u3} ι S) (Function.onFun.{succ u3, succ u2, 1} ι R Prop (IsCoprime.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) q)) -> (Module.IsTorsionBy.{u2, u1} R M (CommRing.toCommSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (Finset.prod.{u2, u3} R ι (CommRing.toCommMonoid.{u2} R _inst_1) S (fun (i : ι) => q i))) -> (DirectSum.IsInternal.{u3, u1, u1} (Subtype.{succ u3} ι (fun (x : ι) => Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) x S)) M (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) (fun (a : Subtype.{succ u3} ι (fun (x : ι) => Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) x S)) (b : Subtype.{succ u3} ι (fun (x : ι) => Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) x S)) => Subtype.instDecidableEqSubtype.{u3} ι (fun (x : ι) => Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) x S) (fun (a : ι) (b : ι) => _inst_4 a b) a b) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) (Submodule.addSubmonoidClass.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) (fun (i : Subtype.{succ u3} ι (fun (x : ι) => Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) x S)) => Submodule.torsionBy.{u2, u1} R M (CommRing.toCommSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (q (Subtype.val.{succ u3} ι (fun (x : ι) => Membership.mem.{u3, u3} ι (Finset.{u3} ι) (Finset.instMembershipFinset.{u3} ι) x S) i))))
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_is_internal Submodule.torsionBy_isInternalₓ'. -/
 /-- If the `q i` are pairwise coprime, a `∏ i, q i`-torsion module is the internal direct sum of
 its `q i`-torsion submodules.-/
 theorem torsionBy_isInternal {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCoprime on q))
     (hM : Module.IsTorsionBy R M <| ∏ i in S, q i) :
     DirectSum.IsInternal fun i : S => torsionBy R M <| q i :=
   by
-  rw [← Module.isTorsionBy_span_singleton_iff, Ideal.submodule_span_eq, ←
+  rw [← Module.isTorsionBySet_span_singleton_iff, Ideal.submodule_span_eq, ←
     Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf] at hM
   convert torsion_by_set_is_internal
       (fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij) hM
@@ -520,8 +722,9 @@ variable {I : Ideal R} (hM : IsTorsionBySet R M I)
 
 include hM
 
+#print Module.IsTorsionBySet.hasSMul /-
 /-- can't be an instance because hM can't be inferred -/
-def IsTorsionBySet.hasSmul : SMul (R ⧸ I) M
+def IsTorsionBySet.hasSMul : SMul (R ⧸ I) M
     where smul b x :=
     Quotient.liftOn' b (· • x) fun b₁ b₂ h =>
       by
@@ -529,8 +732,15 @@ def IsTorsionBySet.hasSmul : SMul (R ⧸ I) M
       have : (-b₁ + b₂) • x = 0 := @hM x ⟨_, quotient_add_group.left_rel_apply.mp h⟩
       rw [add_smul, neg_smul, neg_add_eq_zero] at this
       exact this
-#align module.is_torsion_by_set.has_smul Module.IsTorsionBySet.hasSmul
+#align module.is_torsion_by_set.has_smul Module.IsTorsionBySet.hasSMul
+-/
 
+/- warning: module.is_torsion_by_set.mk_smul -> Module.IsTorsionBySet.mk_smul is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} (hM : Module.IsTorsionBySet.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R 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R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (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_2))))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) b x)
+Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_set.mk_smul Module.IsTorsionBySet.mk_smulₓ'. -/
 @[simp]
 theorem IsTorsionBySet.mk_smul (b : R) (x : M) :
     haveI := hM.has_smul
@@ -538,12 +748,20 @@ theorem IsTorsionBySet.mk_smul (b : R) (x : M) :
   rfl
 #align module.is_torsion_by_set.mk_smul Module.IsTorsionBySet.mk_smul
 
+#print Module.IsTorsionBySet.module /-
 /-- A `(R ⧸ I)`-module is a `R`-module which `is_torsion_by_set R M I`. -/
 def IsTorsionBySet.module : Module (R ⧸ I) M :=
   @Function.Surjective.moduleLeft _ _ _ _ _ _ _ hM.SMul _ Ideal.Quotient.mk_surjective
     (IsTorsionBySet.mk_smul hM)
 #align module.is_torsion_by_set.module Module.IsTorsionBySet.module
+-/
 
+/- warning: module.is_torsion_by_set.is_scalar_tower -> Module.IsTorsionBySet.isScalarTower is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} (hM : Module.IsTorsionBySet.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{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)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) I)) {S : Type.{u3}} [_inst_4 : SMul.{u3, u1} S R] [_inst_5 : SMul.{u3, u2} S M] [_inst_6 : IsScalarTower.{u3, u1, u2} S R 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_2)))) (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_2)))) (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_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) _inst_5] [_inst_7 : IsScalarTower.{u3, u1, u1} S R R _inst_4 (Mul.toSMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) _inst_4], IsScalarTower.{u3, u1, u2} S (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) M (Submodule.Quotient.hasSmul'.{u1, u1, u3} 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))) S _inst_4 _inst_4 _inst_7 I) (MulAction.toHasSmul.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) M (MonoidWithZero.toMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Semiring.toMonoidWithZero.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (DistribMulAction.toMulAction.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) M (MonoidWithZero.toMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Semiring.toMonoidWithZero.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)) (Module.toDistribMulAction.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) M (Ring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (Module.IsTorsionBySet.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 I hM)))) _inst_5
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] {I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} (hM : Module.IsTorsionBySet.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (SetLike.coe.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I)) {S : Type.{u3}} [_inst_4 : SMul.{u3, u1} S R] [_inst_5 : SMul.{u3, u2} S M] [_inst_6 : IsScalarTower.{u3, u1, u2} S R 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_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{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_2))))) (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_2))))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) _inst_5] [_inst_7 : IsScalarTower.{u3, u1, u1} S R R _inst_4 (Algebra.toSMul.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _inst_4], IsScalarTower.{u3, u1, u2} S (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) M (Submodule.Quotient.hasSmul'.{u1, u1, u3} R R (CommRing.toRing.{u1} R _inst_1) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) S _inst_4 _inst_4 _inst_7 I) (MulAction.toSMul.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) M (MonoidWithZero.toMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Semiring.toMonoidWithZero.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (DistribMulAction.toMulAction.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) M (MonoidWithZero.toMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Semiring.toMonoidWithZero.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)) (Module.toDistribMulAction.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) M (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (Module.IsTorsionBySet.module.{u1, u2} R M _inst_1 _inst_2 _inst_3 I hM)))) _inst_5
+Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_set.is_scalar_tower Module.IsTorsionBySet.isScalarTowerₓ'. -/
 instance IsTorsionBySet.isScalarTower {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M]
     [IsScalarTower S R R] : @IsScalarTower S (R ⧸ I) M _ (IsTorsionBySet.module hM).toSMul _
     where smul_assoc b d x := Quotient.inductionOn' d fun c => (smul_assoc b c x : _)
@@ -565,6 +783,12 @@ namespace Submodule
 instance (I : Ideal R) : Module (R ⧸ I) (torsionBySet R M I) :=
   Module.IsTorsionBySet.module <| torsionBySet_isTorsionBySet I
 
+/- warning: submodule.torsion_by_set.mk_smul -> Submodule.torsionBySet.mk_smul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (b : R) (x : coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.torsionBySet.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) 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(HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (fun (_x : RingHom.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) => R -> (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I)) (RingHom.hasCoeToFun.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) I) (Ideal.Quotient.commRing.{u1} R _inst_1 I))))) (Ideal.Quotient.mk.{u1} R _inst_1 I) b) x) (SMul.smul.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.torsionBySet.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{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)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) I))) (Submodule.smul.{u1, u1, u2} R R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.torsionBySet.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{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)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) I)) (Mul.toSMul.{u1} R (MulOneClass.toHasMul.{u1} R (Monoid.toMulOneClass.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (MulAction.toHasSmul.{u1, u2} R M (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) (MulActionWithZero.toMulAction.{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_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (IsScalarTower.left.{u1, u2} R M (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) (MulActionWithZero.toMulAction.{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_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) b x)
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : AddCommGroup.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2)] (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (b : R) (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_2) _inst_3) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3)) x (Submodule.torsionBySet.{u2, u1} R M (CommRing.toCommSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (SetLike.coe.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) I)))), Eq.{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_2) _inst_3) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3)) x (Submodule.torsionBySet.{u2, u1} R M (CommRing.toCommSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (SetLike.coe.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) I)))) (HSMul.hSMul.{u2, u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) b) (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_2) _inst_3) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3)) x (Submodule.torsionBySet.{u2, u1} R M (CommRing.toCommSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (SetLike.coe.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R 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_inst_3)) x (Submodule.torsionBySet.{u2, u1} R M (CommRing.toCommSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (SetLike.coe.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) I)))) (instHSMul.{u2, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) b) (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_2) _inst_3) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3)) x (Submodule.torsionBySet.{u2, u1} R M (CommRing.toCommSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (SetLike.coe.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) I)))) (SMulZeroClass.toSMul.{u2, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) b) (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_2) _inst_3) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} 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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3)) x (Submodule.torsionBySet.{u2, u1} R M (CommRing.toCommSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (SetLike.coe.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} 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(Semiring.toNonAssocSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommSemiring.toSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} R _inst_1) I) (Ideal.Quotient.commRing.{u2} R _inst_1 I)))))))) (Ideal.Quotient.mk.{u2} R _inst_1 I) b) x) (HSMul.hSMul.{u2, u1, u1} 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_2) 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(CommRing.toCommSemiring.{u2} R _inst_1)))) I)))) (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_2) _inst_3) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3) M (Submodule.setLike.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3)) x (Submodule.torsionBySet.{u2, u1} R M (CommRing.toCommSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3 (SetLike.coe.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) 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(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_2))))) (Module.toMulActionWithZero.{u2, u1} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} M _inst_2) _inst_3))))) b x)
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by_set.mk_smul Submodule.torsionBySet.mk_smulₓ'. -/
 @[simp]
 theorem torsionBySet.mk_smul (I : Ideal R) (b : R) (x : torsionBySet R M I) :
     Ideal.Quotient.mk I b • x = b • x :=
@@ -578,8 +802,14 @@ instance (I : Ideal R) {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M]
 /-- The `a`-torsion submodule as a `(R ⧸ R∙a)`-module. -/
 instance (a : R) : Module (R ⧸ R ∙ a) (torsionBy R M a) :=
   Module.IsTorsionBySet.module <|
-    (Module.isTorsionBy_span_singleton_iff a).mpr <| torsionBy_isTorsionBy a
-
+    (Module.isTorsionBySet_span_singleton_iff a).mpr <| torsionBy_isTorsionBy a
+
+/- warning: submodule.torsion_by.mk_smul -> Submodule.torsionBy.mk_smul is a dubious translation:
+lean 3 declaration is
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(Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) a))))))) => R -> (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) a)))) (RingHom.hasCoeToFun.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) a))) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) a))) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) a))) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) a))) (Ideal.Quotient.commRing.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) a))))))) (Ideal.Quotient.mk.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) a))) b) x) (SMul.smul.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a)) (Submodule.smul.{u1, u1, u2} R R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a) (Mul.toSMul.{u1} R (MulOneClass.toHasMul.{u1} R (Monoid.toMulOneClass.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (MulAction.toHasSmul.{u1, u2} R M (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) (MulActionWithZero.toMulAction.{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_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (IsScalarTower.left.{u1, u2} R M (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) (MulActionWithZero.toMulAction.{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_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) b x)
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (a : R) (b : R) (x : 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_2) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) x (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a))), Eq.{succ u2} (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_2) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) x (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a))) (HSMul.hSMul.{u1, u2, u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) b) (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_2) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) x (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a))) (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_2) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) x (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a))) (instHSMul.{u1, u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => HasQuotient.Quotient.{u1, u1} 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_inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) x (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a))) (SMulZeroClass.toSMul.{u1, u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (Ideal.Quotient.commRing.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))))))) R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R 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(CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (Ideal.Quotient.commRing.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))))))) (RingHomClass.toNonUnitalRingHomClass.{u1, u1, u1} (RingHom.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (Ideal.Quotient.commRing.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))))))) R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (Ideal.Quotient.commRing.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a)))))) (RingHom.instRingHomClassRingHom.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) (Ideal.Quotient.commRing.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a)))))))))) (Ideal.Quotient.mk.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) a))) b) x) (HSMul.hSMul.{u1, u2, u2} 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_2) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) x (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a))) (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_2) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) x (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a))) (instHSMul.{u1, u2} 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_2) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) x (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a))) (Submodule.smul.{u1, u1, u2} R R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 (Submodule.torsionBy.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3 a) (Algebra.toSMul.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (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_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{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_2))))) (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_2))))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))) (IsScalarTower.left.{u1, u2} R M (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (MulActionWithZero.toMulAction.{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_2))))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) b x)
+Case conversion may be inaccurate. Consider using '#align submodule.torsion_by.mk_smul Submodule.torsionBy.mk_smulₓ'. -/
 @[simp]
 theorem torsionBy.mk_smul (a b : R) (x : torsionBy R M a) :
     Ideal.Quotient.mk (R ∙ a) b • x = b • x :=
@@ -604,11 +834,23 @@ variable [CommSemiring R] [AddCommMonoid M] [Module R M]
 
 variable (S : Type _) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S R M]
 
+/- warning: submodule.mem_torsion'_iff -> Submodule.mem_torsion'_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (S : Type.{u3}) [_inst_4 : CommMonoid.{u3} S] [_inst_5 : DistribMulAction.{u3, u2} S M (CommMonoid.toMonoid.{u3} S _inst_4) (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] [_inst_6 : SMulCommClass.{u3, u1, u2} S R M (SMulZeroClass.toHasSmul.{u3, u2} S M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u3, u2} S M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u3, u2} S M (CommMonoid.toMonoid.{u3} S _inst_4) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_5))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} 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.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))))] (x : M), Iff (Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsion'.{u1, u2, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6)) (Exists.{succ u3} S (fun (a : S) => Eq.{succ u2} M (SMul.smul.{u3, u2} S M (SMulZeroClass.toHasSmul.{u3, u2} S M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u3, u2} S M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u3, u2} S M (CommMonoid.toMonoid.{u3} S _inst_4) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_5))) a x) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))))))))
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] (S : Type.{u1}) [_inst_4 : CommMonoid.{u1} S] [_inst_5 : DistribMulAction.{u1, u3} S M (CommMonoid.toMonoid.{u1} S _inst_4) (AddCommMonoid.toAddMonoid.{u3} M _inst_2)] [_inst_6 : SMulCommClass.{u1, u2, u3} S R M (SMulZeroClass.toSMul.{u1, u3} S M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u3} S M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u3} S M (CommMonoid.toMonoid.{u1} S _inst_4) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_5))) (SMulZeroClass.toSMul.{u2, u3} R M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R _inst_1)) (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Module.toMulActionWithZero.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3))))] (x : M), Iff (Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsion'.{u1, u2, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6)) (Exists.{succ u1} S (fun (a : S) => Eq.{succ u3} M (HSMul.hSMul.{u1, u3, u3} S M M (instHSMul.{u1, u3} S M (SMulZeroClass.toSMul.{u1, u3} S M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u3} S M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u3} S M (CommMonoid.toMonoid.{u1} S _inst_4) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_5)))) a x) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))))))
+Case conversion may be inaccurate. Consider using '#align submodule.mem_torsion'_iff Submodule.mem_torsion'_iffₓ'. -/
 @[simp]
 theorem mem_torsion'_iff (x : M) : x ∈ torsion' R M S ↔ ∃ a : S, a • x = 0 :=
   Iff.rfl
 #align submodule.mem_torsion'_iff Submodule.mem_torsion'_iff
 
+/- warning: submodule.mem_torsion_iff -> Submodule.mem_torsion_iff 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 submodule.mem_torsion_iff Submodule.mem_torsion_iffₓ'. -/
 @[simp]
 theorem mem_torsion_iff (x : M) : x ∈ torsion R M ↔ ∃ a : R⁰, a • x = 0 :=
   Iff.rfl
@@ -630,6 +872,12 @@ instance : DistribMulAction S (torsion' R M S) :=
 instance : SMulCommClass S R (torsion' R M S) :=
   ⟨fun s a x => Subtype.ext <| smul_comm _ _ _⟩
 
+/- warning: submodule.is_torsion'_iff_torsion'_eq_top -> Submodule.isTorsion'_iff_torsion'_eq_top is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (S : Type.{u3}) [_inst_4 : CommMonoid.{u3} S] [_inst_5 : DistribMulAction.{u3, u2} S M (CommMonoid.toMonoid.{u3} S _inst_4) (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] [_inst_6 : SMulCommClass.{u3, u1, u2} S R M (SMulZeroClass.toHasSmul.{u3, u2} S M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u3, u2} S M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u3, u2} S M (CommMonoid.toMonoid.{u3} S _inst_4) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_5))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} 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.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))))], Iff (Module.IsTorsion'.{u2, u3} M _inst_2 S (SMulZeroClass.toHasSmul.{u3, u2} S M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u3, u2} S M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u3, u2} S M (CommMonoid.toMonoid.{u3} S _inst_4) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_5)))) (Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion'.{u1, u2, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6) (Top.top.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.hasTop.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u3}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : Module.{u1, u3} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] (S : Type.{u2}) [_inst_4 : CommMonoid.{u2} S] [_inst_5 : DistribMulAction.{u2, u3} S M (CommMonoid.toMonoid.{u2} S _inst_4) (AddCommMonoid.toAddMonoid.{u3} M _inst_2)] [_inst_6 : SMulCommClass.{u2, u1, u3} S R M (SMulZeroClass.toSMul.{u2, u3} S M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (DistribSMul.toSMulZeroClass.{u2, u3} S M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (DistribMulAction.toDistribSMul.{u2, u3} S M (CommMonoid.toMonoid.{u2} S _inst_4) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_5))) (SMulZeroClass.toSMul.{u1, u3} R M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u1, u3} R M (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u1, u3} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Module.toMulActionWithZero.{u1, u3} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))))], Iff (Module.IsTorsion'.{u3, u2} M _inst_2 S (SMulZeroClass.toSMul.{u2, u3} S M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (DistribSMul.toSMulZeroClass.{u2, u3} S M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (DistribMulAction.toDistribSMul.{u2, u3} S M (CommMonoid.toMonoid.{u2} S _inst_4) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_5)))) (Eq.{succ u3} (Submodule.{u1, u3} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion'.{u2, u1, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6) (Top.top.{u3} (Submodule.{u1, u3} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.instTopSubmodule.{u1, u3} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)))
+Case conversion may be inaccurate. Consider using '#align submodule.is_torsion'_iff_torsion'_eq_top Submodule.isTorsion'_iff_torsion'_eq_topₓ'. -/
 /-- A `S`-torsion module is a module whose `S`-torsion submodule is the full space. -/
 theorem isTorsion'_iff_torsion'_eq_top : IsTorsion' M S ↔ torsion' R M S = ⊤ :=
   ⟨fun h => eq_top_iff.mpr fun _ _ => @h _, fun h x =>
@@ -638,15 +886,33 @@ theorem isTorsion'_iff_torsion'_eq_top : IsTorsion' M S ↔ torsion' R M S = ⊤
     trivial⟩
 #align submodule.is_torsion'_iff_torsion'_eq_top Submodule.isTorsion'_iff_torsion'_eq_top
 
+/- warning: submodule.torsion'_is_torsion' -> Submodule.torsion'_isTorsion' 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 submodule.torsion'_is_torsion' Submodule.torsion'_isTorsion'ₓ'. -/
 /-- The `S`-torsion submodule is a `S`-torsion module. -/
 theorem torsion'_isTorsion' : IsTorsion' (torsion' R M S) S := fun ⟨x, ⟨a, h⟩⟩ => ⟨a, Subtype.ext h⟩
 #align submodule.torsion'_is_torsion' Submodule.torsion'_isTorsion'
 
+/- warning: submodule.torsion'_torsion'_eq_top -> Submodule.torsion'_torsion'_eq_top is a dubious translation:
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+but is expected to have type
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(AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Module.toMulActionWithZero.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3))))], Eq.{succ u3} (Submodule.{u2, u3} R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsion'.{u1, u2, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6))) (CommSemiring.toSemiring.{u2} R _inst_1) (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3 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(Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsion'.{u1, u2, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6))) (CommSemiring.toSemiring.{u2} R _inst_1) (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3 (Submodule.torsion'.{u1, u2, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6)) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3 (Submodule.torsion'.{u1, u2, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6))) (Submodule.instTopSubmodule.{u2, u3} R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3)) x (Submodule.torsion'.{u1, u2, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6))) (CommSemiring.toSemiring.{u2} R _inst_1) (Submodule.addCommMonoid.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3 (Submodule.torsion'.{u1, u2, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6)) (Submodule.module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3 (Submodule.torsion'.{u1, u2, u3} R M _inst_1 _inst_2 _inst_3 S _inst_4 _inst_5 _inst_6))))
+Case conversion may be inaccurate. Consider using '#align submodule.torsion'_torsion'_eq_top Submodule.torsion'_torsion'_eq_topₓ'. -/
 @[simp]
 theorem torsion'_torsion'_eq_top : torsion' R (torsion' R M S) S = ⊤ :=
   (isTorsion'_iff_torsion'_eq_top S).mp <| torsion'_isTorsion' S
 #align submodule.torsion'_torsion'_eq_top Submodule.torsion'_torsion'_eq_top
 
+/- warning: submodule.torsion_torsion_eq_top -> Submodule.torsion_torsion_eq_top is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align submodule.torsion_torsion_eq_top Submodule.torsion_torsion_eq_topₓ'. -/
 /-- The torsion submodule of the torsion submodule (viewed as a module) is the full
 torsion module. -/
 @[simp]
@@ -654,6 +920,12 @@ theorem torsion_torsion_eq_top : torsion R (torsion R M) = ⊤ :=
   torsion'_torsion'_eq_top R⁰
 #align submodule.torsion_torsion_eq_top Submodule.torsion_torsion_eq_top
 
+/- warning: submodule.torsion_is_torsion -> Submodule.torsion_isTorsion 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 submodule.torsion_is_torsion Submodule.torsion_isTorsionₓ'. -/
 /-- The torsion submodule is always a torsion module. -/
 theorem torsion_isTorsion : Module.IsTorsion R (torsion R M) :=
   torsion'_isTorsion' R⁰
@@ -669,6 +941,12 @@ open BigOperators
 
 variable (R M)
 
+/- warning: module.is_torsion_by_set_annihilator_top -> Module.isTorsionBySet_annihilator_top is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) (M : Type.{u2}) [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2], Module.IsTorsionBySet.{u1, u2} R M _inst_1 _inst_2 _inst_3 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (Submodule.annihilator.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Top.top.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.hasTop.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))))
+but is expected to have type
+  forall (R : Type.{u2}) (M : Type.{u1}) [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2], Module.IsTorsionBySet.{u2, u1} R M _inst_1 _inst_2 _inst_3 (SetLike.coe.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) (Submodule.annihilator.{u2, u1} R M _inst_1 _inst_2 _inst_3 (Top.top.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Submodule.instTopSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3))))
+Case conversion may be inaccurate. Consider using '#align module.is_torsion_by_set_annihilator_top Module.isTorsionBySet_annihilator_topₓ'. -/
 theorem Module.isTorsionBySet_annihilator_top :
     Module.IsTorsionBySet R M (⊤ : Submodule R M).annihilator := fun x ha =>
   mem_annihilator.mp ha.Prop x mem_top
@@ -676,6 +954,12 @@ theorem Module.isTorsionBySet_annihilator_top :
 
 variable {R M}
 
+/- warning: submodule.annihilator_top_inter_non_zero_divisors -> Submodule.annihilator_top_inter_nonZeroDivisors is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_4 : Module.Finite.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3], (Module.IsTorsion.{u1, u2} R M _inst_1 _inst_2 _inst_3) -> (Ne.{succ u1} (Set.{u1} R) (Inter.inter.{u1} (Set.{u1} R) (Set.hasInter.{u1} R) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) R (Submodule.setLike.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (Submodule.annihilator.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Top.top.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.hasTop.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Set.{u1} R) (HasLiftT.mk.{succ u1, succ u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Set.{u1} R) (CoeTCₓ.coe.{succ u1, succ u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (Set.{u1} R) (SetLike.Set.hasCoeT.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (EmptyCollection.emptyCollection.{u1} (Set.{u1} R) (Set.hasEmptyc.{u1} R)))
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] [_inst_4 : Module.Finite.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3], (Module.IsTorsion.{u2, u1} R M _inst_1 _inst_2 _inst_3) -> (Ne.{succ u2} (Set.{u2} R) (Inter.inter.{u2} (Set.{u2} R) (Set.instInterSet.{u2} R) (SetLike.coe.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) (Submodule.annihilator.{u2, u1} R M _inst_1 _inst_2 _inst_3 (Top.top.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Submodule.instTopSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3)))) (SetLike.coe.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (MonoidWithZero.toMulZeroOneClass.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))) (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))))) (EmptyCollection.emptyCollection.{u2} (Set.{u2} R) (Set.instEmptyCollectionSet.{u2} R)))
+Case conversion may be inaccurate. Consider using '#align submodule.annihilator_top_inter_non_zero_divisors Submodule.annihilator_top_inter_nonZeroDivisorsₓ'. -/
 theorem Submodule.annihilator_top_inter_nonZeroDivisors [Module.Finite R M]
     (hM : Module.IsTorsion R M) : ((⊤ : Submodule R M).annihilator : Set R) ∩ R⁰ ≠ ∅ :=
   by
@@ -690,6 +974,12 @@ theorem Submodule.annihilator_top_inter_nonZeroDivisors [Module.Finite R M]
 
 variable [NoZeroDivisors R] [Nontrivial R]
 
+/- warning: submodule.coe_torsion_eq_annihilator_ne_bot -> Submodule.coe_torsion_eq_annihilator_ne_bot is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_4 : NoZeroDivisors.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))] [_inst_5 : Nontrivial.{u1} R], Eq.{succ u2} (Set.{u2} M) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Set.{u2} M) (CoeTCₓ.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)))) (Submodule.torsion.{u1, u2} R M _inst_1 _inst_2 _inst_3)) (setOf.{u2} M (fun (x : M) => Ne.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.annihilator.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Submodule.span.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.hasSingleton.{u2} M) x))) (Bot.bot.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.hasBot.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_4 : NoZeroDivisors.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1))] [_inst_5 : Nontrivial.{u1} R], Eq.{succ u2} (Set.{u2} M) (SetLike.coe.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M _inst_1 _inst_2 _inst_3)) (setOf.{u2} M (fun (x : M) => Ne.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.annihilator.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Submodule.span.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.instSingletonSet.{u2} M) x))) (Bot.bot.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Submodule.instBotSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))))
+Case conversion may be inaccurate. Consider using '#align submodule.coe_torsion_eq_annihilator_ne_bot Submodule.coe_torsion_eq_annihilator_ne_botₓ'. -/
 theorem coe_torsion_eq_annihilator_ne_bot :
     (torsion R M : Set M) = { x : M | (R ∙ x).annihilator ≠ ⊥ } :=
   by
@@ -701,6 +991,12 @@ theorem coe_torsion_eq_annihilator_ne_bot :
       fun ⟨a, hax, ha⟩ => ⟨⟨_, mem_nonZeroDivisors_of_ne_zero ha⟩, hax x ⟨1, one_smul _ _⟩⟩⟩
 #align submodule.coe_torsion_eq_annihilator_ne_bot Submodule.coe_torsion_eq_annihilator_ne_bot
 
+/- warning: submodule.no_zero_smul_divisors_iff_torsion_eq_bot -> Submodule.noZeroSMulDivisors_iff_torsion_eq_bot is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_4 : NoZeroDivisors.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))] [_inst_5 : Nontrivial.{u1} R], Iff (NoZeroSMulDivisors.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} 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.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))))) (Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M _inst_1 _inst_2 _inst_3) (Bot.bot.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.hasBot.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)))
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u1}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u1} M] [_inst_3 : Module.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] [_inst_4 : NoZeroDivisors.{u2} R (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R _inst_1))] [_inst_5 : Nontrivial.{u2} R], Iff (NoZeroSMulDivisors.{u2, u1} R M (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R _inst_1)) (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2)) (SMulZeroClass.toSMul.{u2, u1} R M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u2, u1} R M (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R _inst_1)) (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u2, u1} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_2)) (Module.toMulActionWithZero.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3))))) (Eq.{succ u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u2, u1} R M _inst_1 _inst_2 _inst_3) (Bot.bot.{u1} (Submodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Submodule.instBotSubmodule.{u2, u1} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_3)))
+Case conversion may be inaccurate. Consider using '#align submodule.no_zero_smul_divisors_iff_torsion_eq_bot Submodule.noZeroSMulDivisors_iff_torsion_eq_botₓ'. -/
 /-- A module over a domain has `no_zero_smul_divisors` iff its torsion submodule is trivial. -/
 theorem noZeroSMulDivisors_iff_torsion_eq_bot : NoZeroSMulDivisors R M ↔ torsion R M = ⊥ :=
   by
@@ -732,6 +1028,12 @@ namespace QuotientTorsion
 
 variable [CommRing R] [AddCommGroup M] [Module R M]
 
+/- warning: submodule.quotient_torsion.torsion_eq_bot -> Submodule.QuotientTorsion.torsion_eq_bot is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)], Eq.{succ u2} (Submodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.Quotient.module.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.torsion.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R 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(Submodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.Quotient.module.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.hasBot.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.Quotient.module.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)], Eq.{succ u2} (Submodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.Quotient.module.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.torsion.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.Quotient.module.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Bot.bot.{u2} (Submodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.Quotient.module.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.instBotSubmodule.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.Quotient.module.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))
+Case conversion may be inaccurate. Consider using '#align submodule.quotient_torsion.torsion_eq_bot Submodule.QuotientTorsion.torsion_eq_botₓ'. -/
 /-- Quotienting by the torsion submodule gives a torsion-free module. -/
 @[simp]
 theorem torsion_eq_bot : torsion R (M ⧸ torsion R M) = ⊥ :=
@@ -744,6 +1046,12 @@ theorem torsion_eq_bot : torsion R (M ⧸ torsion R M) = ⊥ :=
       exact ⟨b * a, (mul_smul _ _ _).trans h⟩
 #align submodule.quotient_torsion.torsion_eq_bot Submodule.QuotientTorsion.torsion_eq_bot
 
+/- warning: submodule.quotient_torsion.no_zero_smul_divisors -> Submodule.QuotientTorsion.noZeroSMulDivisors is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_4 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))], NoZeroSMulDivisors.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (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} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddMonoid.toAddZeroClass.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddCommMonoid.toAddMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R 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(SMulZeroClass.toHasSmul.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddZeroClass.toHasZero.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddMonoid.toAddZeroClass.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddCommMonoid.toAddMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddMonoid.toAddZeroClass.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddCommMonoid.toAddMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddMonoid.toAddZeroClass.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddCommMonoid.toAddMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) (Module.toMulActionWithZero.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.addCommGroup.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))) (Submodule.Quotient.module.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_4 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))], NoZeroSMulDivisors.{u1, u2} R (HasQuotient.Quotient.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_4))) (Submodule.Quotient.instZeroQuotientSubmoduleToSemiringToAddCommMonoidHasQuotient.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)) (Submodule.Quotient.hasSmul.{u1, u2} R M (CommRing.toRing.{u1} R _inst_1) _inst_2 _inst_3 (Submodule.torsion.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))
+Case conversion may be inaccurate. Consider using '#align submodule.quotient_torsion.no_zero_smul_divisors Submodule.QuotientTorsion.noZeroSMulDivisorsₓ'. -/
 instance noZeroSMulDivisors [IsDomain R] : NoZeroSMulDivisors R (M ⧸ torsion R M) :=
   noZeroSMulDivisors_iff_torsion_eq_bot.mpr torsion_eq_bot
 #align submodule.quotient_torsion.no_zero_smul_divisors Submodule.QuotientTorsion.noZeroSMulDivisors
@@ -758,6 +1066,12 @@ section
 
 variable [Monoid R] [AddCommMonoid M] [DistribMulAction R M]
 
+/- warning: submodule.is_torsion'_powers_iff -> Submodule.isTorsion'_powers_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Monoid.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : DistribMulAction.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] (p : R), Iff (Module.IsTorsion'.{u2, u1} M _inst_2 (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) R (Submonoid.setLike.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1))) (Submonoid.powers.{u1} R _inst_1 p)) (Submonoid.hasSmul.{u1, u2} R M (Monoid.toMulOneClass.{u1} R _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3))) (Submonoid.powers.{u1} R _inst_1 p))) (forall (x : M), Exists.{1} Nat (fun (n : Nat) => Eq.{succ u2} M (SMul.smul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R _inst_1)) p n) x) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))))))))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Monoid.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : DistribMulAction.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] (p : R), Iff (Module.IsTorsion'.{u2, u1} M _inst_2 (Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) R (Submonoid.instSetLikeSubmonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1))) x (Submonoid.powers.{u1} R _inst_1 p))) (Submonoid.smul.{u1, u2} R M (Monoid.toMulOneClass.{u1} R _inst_1) (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3))) (Submonoid.powers.{u1} R _inst_1 p))) (forall (x : M), Exists.{1} Nat (fun (n : Nat) => Eq.{succ u2} M (HSMul.hSMul.{u1, u2, u2} R M M (instHSMul.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3)))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R _inst_1)) p n) x) (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))))))
+Case conversion may be inaccurate. Consider using '#align submodule.is_torsion'_powers_iff Submodule.isTorsion'_powers_iffₓ'. -/
 theorem isTorsion'_powers_iff (p : R) :
     IsTorsion' M (Submonoid.powers p) ↔ ∀ x : M, ∃ n : ℕ, p ^ n • x = 0 :=
   ⟨fun h x =>
@@ -768,6 +1082,12 @@ theorem isTorsion'_powers_iff (p : R) :
     ⟨⟨_, ⟨n, rfl⟩⟩, hn⟩⟩
 #align submodule.is_torsion'_powers_iff Submodule.isTorsion'_powers_iff
 
+/- warning: submodule.p_order -> Submodule.pOrder is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Monoid.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : DistribMulAction.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] {p : R}, (Module.IsTorsion'.{u2, u1} M _inst_2 (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) R (Submonoid.setLike.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1))) (Submonoid.powers.{u1} R _inst_1 p)) (Submonoid.hasSmul.{u1, u2} R M (Monoid.toMulOneClass.{u1} R _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3))) (Submonoid.powers.{u1} R _inst_1 p))) -> (forall (x : M) [_inst_4 : forall (n : Nat), Decidable (Eq.{succ u2} M (SMul.smul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R _inst_1)) p n) x) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))))))], Nat)
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Monoid.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : DistribMulAction.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] {p : R}, (Module.IsTorsion'.{u2, u1} M _inst_2 (Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) R (Submonoid.instSetLikeSubmonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1))) x (Submonoid.powers.{u1} R _inst_1 p))) (Submonoid.smul.{u1, u2} R M (Monoid.toMulOneClass.{u1} R _inst_1) (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3))) (Submonoid.powers.{u1} R _inst_1 p))) -> (forall (x : M) [_inst_4 : forall (n : Nat), Decidable (Eq.{succ u2} M (HSMul.hSMul.{u1, u2, u2} R M M (instHSMul.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3)))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R _inst_1)) p n) x) (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))))], Nat)
+Case conversion may be inaccurate. Consider using '#align submodule.p_order Submodule.pOrderₓ'. -/
 /-- In a `p ^ ∞`-torsion module (that is, a module where all elements are cancelled by scalar
 multiplication by some power of `p`), the smallest `n` such that `p ^ n • x = 0`.-/
 def pOrder {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (x : M)
@@ -775,6 +1095,12 @@ def pOrder {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (x : M)
   Nat.find <| (isTorsion'_powers_iff p).mp hM x
 #align submodule.p_order Submodule.pOrder
 
+/- warning: submodule.pow_p_order_smul -> Submodule.pow_pOrder_smul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Monoid.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : DistribMulAction.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] {p : R} (hM : Module.IsTorsion'.{u2, u1} M _inst_2 (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) R (Submonoid.setLike.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1))) (Submonoid.powers.{u1} R _inst_1 p)) (Submonoid.hasSmul.{u1, u2} R M (Monoid.toMulOneClass.{u1} R _inst_1) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3))) (Submonoid.powers.{u1} R _inst_1 p))) (x : M) [_inst_4 : forall (n : Nat), Decidable (Eq.{succ u2} M (SMul.smul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R _inst_1)) p n) x) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))))))], Eq.{succ u2} M (SMul.smul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R _inst_1)) p (Submodule.pOrder.{u1, u2} R M _inst_1 _inst_2 _inst_3 p hM x (fun (n : Nat) => _inst_4 n))) x) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))))))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Monoid.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : DistribMulAction.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] {p : R} (hM : Module.IsTorsion'.{u2, u1} M _inst_2 (Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1)) R (Submonoid.instSetLikeSubmonoid.{u1} R (Monoid.toMulOneClass.{u1} R _inst_1))) x (Submonoid.powers.{u1} R _inst_1 p))) (Submonoid.smul.{u1, u2} R M (Monoid.toMulOneClass.{u1} R _inst_1) (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3))) (Submonoid.powers.{u1} R _inst_1 p))) (x : M) [_inst_4 : forall (n : Nat), Decidable (Eq.{succ u2} M (HSMul.hSMul.{u1, u2, u2} R M M (instHSMul.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3)))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R _inst_1)) p n) x) (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))))], Eq.{succ u2} M (HSMul.hSMul.{u1, u2, u2} R M M (instHSMul.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M _inst_1 (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_3)))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R _inst_1)) p (Submodule.pOrder.{u1, u2} R M _inst_1 _inst_2 _inst_3 p hM x (fun (n : Nat) => _inst_4 n))) x) (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))))
+Case conversion may be inaccurate. Consider using '#align submodule.pow_p_order_smul Submodule.pow_pOrder_smulₓ'. -/
 @[simp]
 theorem pow_pOrder_smul {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (x : M)
     [∀ n : ℕ, Decidable (p ^ n • x = 0)] : p ^ pOrder hM x • x = 0 :=
@@ -785,6 +1111,12 @@ end
 
 variable [CommSemiring R] [AddCommMonoid M] [Module R M] [∀ x : M, Decidable (x = 0)]
 
+/- warning: submodule.exists_is_torsion_by -> Submodule.exists_isTorsionBy is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_4 : forall (x : M), Decidable (Eq.{succ u2} M x (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))))))] {p : R} (hM : Module.IsTorsion'.{u2, u1} M _inst_2 (coeSort.{succ u1, succ (succ u1)} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.setLike.{u1} R (Monoid.toMulOneClass.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (Submonoid.powers.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) p)) (Submonoid.hasSmul.{u1, u2} R M (Monoid.toMulOneClass.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} 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.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)))) (Submonoid.powers.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) p))) (d : Nat), (Ne.{1} Nat d (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (forall (s : (Fin d) -> M), (Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.span.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Set.range.{u2, 1} M (Fin d) s)) (Top.top.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.hasTop.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) -> (Exists.{1} (Fin d) (fun (j : Fin d) => Module.IsTorsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) p (Submodule.pOrder.{u1, u2} R M (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) _inst_2 (Module.toDistribMulAction.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) p hM (s j) (fun (n : Nat) => _inst_4 (SMul.smul.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (DistribSMul.toSmulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) (Module.toDistribMulAction.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) p n) (s j))))))))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_4 : forall (x : M), Decidable (Eq.{succ u2} M x (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))))] {p : R} (hM : Module.IsTorsion'.{u2, u1} M _inst_2 (Subtype.{succ u1} R (fun (x : R) => Membership.mem.{u1, u1} R (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (Monoid.toMulOneClass.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (Monoid.toMulOneClass.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x (Submonoid.powers.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) p))) (Submonoid.smul.{u1, u2} R M (Monoid.toMulOneClass.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3)))) (Submonoid.powers.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) p))) (d : Nat), (Ne.{1} Nat d (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (forall (s : (Fin d) -> M), (Eq.{succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.span.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3 (Set.range.{u2, 1} M (Fin d) s)) (Top.top.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Submodule.instTopSubmodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))) -> (Exists.{1} (Fin d) (fun (j : Fin d) => Module.IsTorsionBy.{u1, u2} R M _inst_1 _inst_2 _inst_3 (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) p (Submodule.pOrder.{u1, u2} R M (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) _inst_2 (Module.toDistribMulAction.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3) p hM (s j) (fun (n : Nat) => _inst_4 (HSMul.hSMul.{u1, u2, u2} R M M (instHSMul.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribSMul.toSMulZeroClass.{u1, u2} R M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (DistribMulAction.toDistribSMul.{u1, u2} R M (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) (Module.toDistribMulAction.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_3))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) p n) (s j))))))))
+Case conversion may be inaccurate. Consider using '#align submodule.exists_is_torsion_by Submodule.exists_isTorsionByₓ'. -/
 theorem exists_isTorsionBy {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (d : ℕ) (hd : d ≠ 0)
     (s : Fin d → M) (hs : span R (Set.range s) = ⊤) :
     ∃ j : Fin d, Module.IsTorsionBy R M (p ^ pOrder hM (s j)) :=
@@ -810,6 +1142,12 @@ namespace Ideal.Quotient
 
 open Submodule
 
+/- warning: ideal.quotient.torsion_by_eq_span_singleton -> Ideal.Quotient.torsionBy_eq_span_singleton is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (a : R) (b : R), (Membership.Mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (SetLike.hasMem.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) a (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Eq.{succ u1} (Submodule.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))) (Submodule.Quotient.addCommGroup.{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))) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b))))) (Submodule.Quotient.module.{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))) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b))))) (Submodule.torsionBy.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R 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(Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))) (Ideal.Quotient.commRing.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))))))) => R -> (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b))))) (RingHom.hasCoeToFun.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))) (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))) (CommRing.toRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))) (Ideal.Quotient.commRing.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))))))) (Ideal.Quotient.mk.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) a b)))) b))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (a : R) (b : R), (Membership.mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))))) a (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) -> (Eq.{succ u1} (Submodule.{u1, u1} R (HasQuotient.Quotient.{u1, u1} R (Submodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) a b)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (HasQuotient.Quotient.{u1, u1} R (Submodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) a b)))) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Submodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) a b)))) (NonAssocRing.toNonUnitalNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Submodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) a b)))) (Ring.toNonAssocRing.{u1} (HasQuotient.Quotient.{u1, u1} R (Submodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R 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(Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) a b)))) (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) a b)))) (CommSemiring.toSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) a b)))) (CommRing.toCommSemiring.{u1} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) a b)))) (Ideal.Quotient.commRing.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) a b))))))))))) (Ideal.Quotient.mk.{u1} R _inst_1 (Submodule.span.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) a b)))) b))))
+Case conversion may be inaccurate. Consider using '#align ideal.quotient.torsion_by_eq_span_singleton Ideal.Quotient.torsionBy_eq_span_singletonₓ'. -/
 theorem torsionBy_eq_span_singleton {R : Type _} [CommRing R] (a b : R) (ha : a ∈ R⁰) :
     torsionBy R (R ⧸ R ∙ a * b) a = R ∙ mk _ b :=
   by
@@ -830,6 +1168,7 @@ end Ideal.Quotient
 
 namespace AddMonoid
 
+#print AddMonoid.isTorsion_iff_isTorsion_nat /-
 theorem isTorsion_iff_isTorsion_nat [AddCommMonoid M] :
     AddMonoid.IsTorsion M ↔ Module.IsTorsion ℕ M :=
   by
@@ -840,7 +1179,14 @@ theorem isTorsion_iff_isTorsion_nat [AddCommMonoid M] :
     obtain ⟨n, hn⟩ := @h x
     refine' ⟨n, Nat.pos_of_ne_zero (nonZeroDivisors.coe_ne_zero _), hn⟩
 #align add_monoid.is_torsion_iff_is_torsion_nat AddMonoid.isTorsion_iff_isTorsion_nat
+-/
 
+/- warning: add_monoid.is_torsion_iff_is_torsion_int -> AddMonoid.isTorsion_iff_isTorsion_int is a dubious translation:
+lean 3 declaration is
+  forall {M : Type.{u1}} [_inst_1 : AddCommGroup.{u1} M], Iff (AddMonoid.IsTorsion.{u1} M (SubNegMonoid.toAddMonoid.{u1} M (AddGroup.toSubNegMonoid.{u1} M (AddCommGroup.toAddGroup.{u1} M _inst_1)))) (Module.IsTorsion.{0, u1} Int M Int.commSemiring (AddCommGroup.toAddCommMonoid.{u1} M _inst_1) (AddCommGroup.intModule.{u1} M _inst_1))
+but is expected to have type
+  forall {M : Type.{u1}} [_inst_1 : AddCommGroup.{u1} M], Iff (AddMonoid.IsTorsion.{u1} M (SubNegMonoid.toAddMonoid.{u1} M (AddGroup.toSubNegMonoid.{u1} M (AddCommGroup.toAddGroup.{u1} M _inst_1)))) (Module.IsTorsion.{0, u1} Int M Int.instCommSemiringInt (AddCommGroup.toAddCommMonoid.{u1} M _inst_1) (AddCommGroup.intModule.{u1} M _inst_1))
+Case conversion may be inaccurate. Consider using '#align add_monoid.is_torsion_iff_is_torsion_int AddMonoid.isTorsion_iff_isTorsion_intₓ'. -/
 theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
     AddMonoid.IsTorsion M ↔ Module.IsTorsion ℤ M :=
   by

mathlib commit https://github.com/leanprover-community/mathlib/commit/e3fb84046afd187b710170887195d50bada934ee

@@ -117,7 +117,7 @@ theorem CompleteLattice.Independent.linear_independent' {ι R M : Type _} {v : 
   by
   refine' linear_independent_iff_not_smul_mem_span.mpr fun i r hi => _
   replace hv := complete_lattice.independent_def.mp hv i
-  simp only [supᵢ_subtype', ← Submodule.span_range_eq_supᵢ, disjoint_iff] at hv
+  simp only [iSup_subtype', ← Submodule.span_range_eq_iSup, disjoint_iff] at hv
   have : r • v i ∈ ⊥ := by
     rw [← hv, Submodule.mem_inf]
     refine' ⟨submodule.mem_span_singleton.mpr ⟨r, rfl⟩, _⟩
@@ -171,7 +171,7 @@ def torsionBy (a : R) : Submodule R M :=
 /-- The submodule containing all elements `x` of `M` such that `a • x = 0` for all `a` in `s`. -/
 @[simps]
 def torsionBySet (s : Set R) : Submodule R M :=
-  infₛ (torsionBy R M '' s)
+  sInf (torsionBy R M '' s)
 #align submodule.torsion_by_set Submodule.torsionBySet
 
 /-- The `S`-torsion submodule, containing all elements `x` of `M` such that `a • x = 0` for some
@@ -266,7 +266,7 @@ theorem torsionBy_singleton_eq : torsionBySet R M {a} = torsionBy R M a :=
 
 theorem torsionBySet_le_torsionBySet_of_subset {s t : Set R} (st : s ⊆ t) :
     torsionBySet R M t ≤ torsionBySet R M s :=
-  infₛ_le_infₛ fun _ ⟨a, ha, h⟩ => ⟨a, st ha, h⟩
+  sInf_le_sInf fun _ ⟨a, ha, h⟩ => ⟨a, st ha, h⟩
 #align submodule.torsion_by_set_le_torsion_by_set_of_subset Submodule.torsionBySet_le_torsionBySet_of_subset
 
 /-- Torsion by a set is torsion by the ideal generated by it. -/
@@ -390,27 +390,27 @@ variable (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
 
 include hp
 
-theorem supᵢ_torsion_by_ideal_eq_torsion_by_infᵢ :
+theorem iSup_torsion_by_ideal_eq_torsion_by_iInf :
     (⨆ i ∈ S, torsionBySet R M <| p i) = torsionBySet R M ↑(⨅ i ∈ S, p i) :=
   by
   cases' S.eq_empty_or_nonempty with h h
   · rw [h]
-    convert supᵢ_emptyset
+    convert iSup_emptyset
     convert torsion_by_univ
     convert top_coe
-    exact infᵢ_emptyset
+    exact iInf_emptyset
   apply le_antisymm
-  · apply supᵢ_le _
+  · apply iSup_le _
     intro i
-    apply supᵢ_le _
+    apply iSup_le _
     intro is
     apply torsion_by_set_le_torsion_by_set_of_subset
-    exact (infᵢ_le (fun i => ⨅ H : i ∈ S, p i) i).trans (infᵢ_le _ is)
+    exact (iInf_le (fun i => ⨅ H : i ∈ S, p i) i).trans (iInf_le _ is)
   · intro x hx
     rw [mem_supr_finset_iff_exists_sum]
     obtain ⟨μ, hμ⟩ :=
       (mem_supr_finset_iff_exists_sum _ _).mp
-        ((Ideal.eq_top_iff_one _).mp <| (Ideal.supᵢ_infᵢ_eq_top_iff_pairwise h _).mpr hp)
+        ((Ideal.eq_top_iff_one _).mp <| (Ideal.iSup_iInf_eq_top_iff_pairwise h _).mpr hp)
     refine' ⟨fun i => ⟨(μ i : R) • x, _⟩, _⟩
     · rw [mem_torsion_by_set_iff] at hx⊢
       rintro ⟨a, ha⟩
@@ -429,17 +429,17 @@ theorem supᵢ_torsion_by_ideal_eq_torsion_by_infᵢ :
         exact Ideal.mul_mem_left _ _ (this j hj ij)
     · simp_rw [coe_mk]
       rw [← Finset.sum_smul, hμ, one_smul]
-#align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.supᵢ_torsion_by_ideal_eq_torsion_by_infᵢ
+#align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsion_by_ideal_eq_torsion_by_iInf
 
 theorem supIndep_torsion_by_ideal : S.SupIndep fun i => torsionBySet R M <| p i :=
   fun T hT i hi hiT =>
   by
-  rw [disjoint_iff, Finset.sup_eq_supᵢ,
+  rw [disjoint_iff, Finset.sup_eq_iSup,
     supr_torsion_by_ideal_eq_torsion_by_infi fun i hi j hj ij => hp (hT hi) (hT hj) ij]
   have :=
     @GaloisConnection.u_inf _ _ (OrderDual.toDual _) (OrderDual.toDual _) _ _ _ _ (torsion_gc R M)
   dsimp at this⊢
-  rw [← this, Ideal.sup_infᵢ_eq_top, top_coe, torsion_by_univ]
+  rw [← this, Ideal.sup_iInf_eq_top, top_coe, torsion_by_univ]
   intro j hj; apply hp hi (hT hj); rintro rfl; exact hiT hj
 #align submodule.sup_indep_torsion_by_ideal Submodule.supIndep_torsion_by_ideal
 
@@ -449,11 +449,11 @@ variable {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCoprime on q))
 
 include hq
 
-theorem supᵢ_torsionBy_eq_torsionBy_prod :
+theorem iSup_torsionBy_eq_torsionBy_prod :
     (⨆ i ∈ S, torsionBy R M <| q i) = torsionBy R M (∏ i in S, q i) :=
   by
   rw [← torsion_by_span_singleton_eq, Ideal.submodule_span_eq, ←
-    Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_infᵢ, ←
+    Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf, ←
     supr_torsion_by_ideal_eq_torsion_by_infi]
   · congr
     ext : 1
@@ -461,7 +461,7 @@ theorem supᵢ_torsionBy_eq_torsionBy_prod :
     ext : 1
     exact (torsion_by_span_singleton_eq _).symm
   · exact fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime _ _).mpr (hq hi hj ij)
-#align submodule.supr_torsion_by_eq_torsion_by_prod Submodule.supᵢ_torsionBy_eq_torsionBy_prod
+#align submodule.supr_torsion_by_eq_torsion_by_prod Submodule.iSup_torsionBy_eq_torsionBy_prod
 
 theorem supIndep_torsionBy : S.SupIndep fun i => torsionBy R M <| q i :=
   by
@@ -492,10 +492,10 @@ theorem torsionBySet_isInternal {p : ι → Ideal R}
     (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
     (hM : Module.IsTorsionBySet R M (⨅ i ∈ S, p i : Ideal R)) :
     DirectSum.IsInternal fun i : S => torsionBySet R M <| p i :=
-  DirectSum.isInternal_submodule_of_independent_of_supᵢ_eq_top
+  DirectSum.isInternal_submodule_of_independent_of_iSup_eq_top
     (CompleteLattice.independent_iff_supIndep.mpr <| supIndep_torsion_by_ideal hp)
-    ((supᵢ_subtype'' ↑S fun i => torsionBySet R M <| p i).trans <|
-      (supᵢ_torsion_by_ideal_eq_torsion_by_infᵢ hp).trans <|
+    ((iSup_subtype'' ↑S fun i => torsionBySet R M <| p i).trans <|
+      (iSup_torsion_by_ideal_eq_torsion_by_iInf hp).trans <|
         (Module.isTorsionBySet_iff_torsionBySet_eq_top _).mp hM)
 #align submodule.torsion_by_set_is_internal Submodule.torsionBySet_isInternal
 
@@ -506,7 +506,7 @@ theorem torsionBy_isInternal {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCo
     DirectSum.IsInternal fun i : S => torsionBy R M <| q i :=
   by
   rw [← Module.isTorsionBy_span_singleton_iff, Ideal.submodule_span_eq, ←
-    Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_infᵢ] at hM
+    Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf] at hM
   convert torsion_by_set_is_internal
       (fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij) hM
   ext : 1; exact (torsion_by_span_singleton_eq _).symm

mathlib commit https://github.com/leanprover-community/mathlib/commit/246f6f7989ff86bd07e1b014846f11304f33cf9e

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Pierre-Alexandre Bazin
 
 ! This file was ported from Lean 3 source module algebra.module.torsion
-! leanprover-community/mathlib commit a50170a88a47570ed186b809ca754110590f9476
+! leanprover-community/mathlib commit cdc34484a07418af43daf8198beaf5c00324bca8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -667,24 +667,26 @@ variable [CommSemiring R] [AddCommMonoid M] [Module R M]
 
 open BigOperators
 
-theorem isTorsion_by_ideal_of_finite_of_isTorsion [Module.Finite R M] (hM : Module.IsTorsion R M) :
-    ∃ I : Ideal R, (I : Set R) ∩ R⁰ ≠ ∅ ∧ Module.IsTorsionBySet R M I :=
+variable (R M)
+
+theorem Module.isTorsionBySet_annihilator_top :
+    Module.IsTorsionBySet R M (⊤ : Submodule R M).annihilator := fun x ha =>
+  mem_annihilator.mp ha.Prop x mem_top
+#align module.is_torsion_by_set_annihilator_top Module.isTorsionBySet_annihilator_top
+
+variable {R M}
+
+theorem Submodule.annihilator_top_inter_nonZeroDivisors [Module.Finite R M]
+    (hM : Module.IsTorsion R M) : ((⊤ : Submodule R M).annihilator : Set R) ∩ R⁰ ≠ ∅ :=
   by
-  cases' (module.finite_def.mp inferInstance : (⊤ : Submodule R M).Fg) with S h
-  refine' ⟨∏ x in S, Ideal.torsionOf R M x, _, _⟩
-  · refine' Set.Nonempty.ne_empty ⟨_, _, (∏ x in S, (@hM x).some : R⁰).2⟩
-    rw [Subtype.val_eq_coe, Submonoid.coe_finset_prod]
-    apply Ideal.prod_mem_prod
-    exact fun x _ => (@hM x).choose_spec
-  · rw [Module.isTorsionBySet_iff_torsionBySet_eq_top, eq_top_iff, ← h, span_le]
-    intro x hx
-    apply torsion_by_set_le_torsion_by_set_of_subset
-    · apply Ideal.le_of_dvd
-      exact Finset.dvd_prod_of_mem _ hx
-    · rw [mem_torsion_by_set_iff]
-      rintro ⟨a, ha⟩
-      exact ha
-#align submodule.is_torsion_by_ideal_of_finite_of_is_torsion Submodule.isTorsion_by_ideal_of_finite_of_isTorsion
+  obtain ⟨S, hS⟩ := ‹Module.Finite R M›.out
+  refine' Set.Nonempty.ne_empty ⟨_, _, (∏ x in S, (@hM x).some : R⁰).Prop⟩
+  rw [Submonoid.coe_finset_prod, SetLike.mem_coe, ← hS, mem_annihilator_span]
+  intro n
+  letI := Classical.decEq M
+  rw [← Finset.prod_erase_mul _ _ n.prop, mul_smul, ← Submonoid.smul_def, (@hM n).choose_spec,
+    smul_zero]
+#align submodule.annihilator_top_inter_non_zero_divisors Submodule.annihilator_top_inter_nonZeroDivisors
 
 variable [NoZeroDivisors R] [Nontrivial R]
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -465,8 +465,7 @@ theorem supᵢ_torsionBy_eq_torsionBy_prod :
 
 theorem supIndep_torsionBy : S.SupIndep fun i => torsionBy R M <| q i :=
   by
-  convert
-    sup_indep_torsion_by_ideal fun i hi j hj ij =>
+  convert sup_indep_torsion_by_ideal fun i hi j hj ij =>
       (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij
   ext : 1; exact (torsion_by_span_singleton_eq _).symm
 #align submodule.sup_indep_torsion_by Submodule.supIndep_torsionBy
@@ -508,8 +507,7 @@ theorem torsionBy_isInternal {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCo
   by
   rw [← Module.isTorsionBy_span_singleton_iff, Ideal.submodule_span_eq, ←
     Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_infᵢ] at hM
-  convert
-    torsion_by_set_is_internal
+  convert torsion_by_set_is_internal
       (fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij) hM
   ext : 1; exact (torsion_by_span_singleton_eq _).symm
 #align submodule.torsion_by_is_internal Submodule.torsionBy_isInternal

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

@@ -834,9 +834,9 @@ theorem isTorsion_iff_isTorsion_nat [AddCommMonoid M] :
     AddMonoid.IsTorsion M ↔ Module.IsTorsion ℕ M :=
   by
   refine' ⟨fun h x => _, fun h x => _⟩
-  · obtain ⟨n, h0, hn⟩ := (is_of_fin_add_order_iff_nsmul_eq_zero x).mp (h x)
+  · obtain ⟨n, h0, hn⟩ := (isOfFinAddOrder_iff_nsmul_eq_zero x).mp (h x)
     exact ⟨⟨n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt h0⟩, hn⟩
-  · rw [is_of_fin_add_order_iff_nsmul_eq_zero]
+  · rw [isOfFinAddOrder_iff_nsmul_eq_zero]
     obtain ⟨n, hn⟩ := @h x
     refine' ⟨n, Nat.pos_of_ne_zero (nonZeroDivisors.coe_ne_zero _), hn⟩
 #align add_monoid.is_torsion_iff_is_torsion_nat AddMonoid.isTorsion_iff_isTorsion_nat
@@ -845,11 +845,11 @@ theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
     AddMonoid.IsTorsion M ↔ Module.IsTorsion ℤ M :=
   by
   refine' ⟨fun h x => _, fun h x => _⟩
-  · obtain ⟨n, h0, hn⟩ := (is_of_fin_add_order_iff_nsmul_eq_zero x).mp (h x)
+  · obtain ⟨n, h0, hn⟩ := (isOfFinAddOrder_iff_nsmul_eq_zero x).mp (h x)
     exact
       ⟨⟨n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt <| int.coe_nat_pos.mpr h0⟩,
         (coe_nat_zsmul _ _).trans hn⟩
-  · rw [is_of_fin_add_order_iff_nsmul_eq_zero]
+  · rw [isOfFinAddOrder_iff_nsmul_eq_zero]
     obtain ⟨n, hn⟩ := @h x
     exact exists_nsmul_eq_zero_of_zsmul_eq_zero (nonZeroDivisors.coe_ne_zero n) hn
 #align add_monoid.is_torsion_iff_is_torsion_int AddMonoid.isTorsion_iff_isTorsion_int

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

This breaks a few simp proofs which were expecting these lemmas to apply to the data binders but not the prop binders.

@@ -112,7 +112,7 @@ theorem CompleteLattice.Independent.linear_independent' {ι R M : Type*} {v : ι
     (h_ne_zero : ∀ i, Ideal.torsionOf R M (v i) = ⊥) : LinearIndependent R v := by
   refine' linearIndependent_iff_not_smul_mem_span.mpr fun i r hi => _
   replace hv := CompleteLattice.independent_def.mp hv i
-  simp only [iSup_subtype', ← Submodule.span_range_eq_iSup, disjoint_iff] at hv
+  simp only [iSup_subtype', ← Submodule.span_range_eq_iSup (ι := Subtype _), disjoint_iff] at hv
   have : r • v i ∈ ⊥ := by
     rw [← hv, Submodule.mem_inf]
     refine' ⟨Submodule.mem_span_singleton.mpr ⟨r, rfl⟩, _⟩

and reduce its scope in a few other instances. Mostly in CategoryTheory and Data this time; some Combinatorics also.

Co-authored-by: Richard Osborn <richardosborn@mac.com>

@@ -170,13 +170,12 @@ def torsionBySet (s : Set R) : Submodule R M :=
   sInf (torsionBy R M '' s)
 #align submodule.torsion_by_set Submodule.torsionBySet
 
-set_option autoImplicit true in
 -- Porting note: torsion' had metavariables and factoring out this fixed it
 -- perhaps there is a better fix
 /-- The additive submonoid of all elements `x` of `M` such that `a • x = 0`
 for some `a` in `S`. -/
 @[simps!]
-def torsion'AddSubMonoid (S : Type w) [CommMonoid S] [DistribMulAction S M] :
+def torsion'AddSubMonoid (S : Type*) [CommMonoid S] [DistribMulAction S M] :
     AddSubmonoid M where
   carrier := { x | ∃ a : S, a • x = 0 }
   add_mem' := by
@@ -185,11 +184,10 @@ def torsion'AddSubMonoid (S : Type w) [CommMonoid S] [DistribMulAction S M] :
     rw [smul_add, mul_smul, mul_comm, mul_smul, hx, hy, smul_zero, smul_zero, add_zero]
   zero_mem' := ⟨1, smul_zero 1⟩
 
-set_option autoImplicit true in
 /-- The `S`-torsion submodule, containing all elements `x` of `M` such that `a • x = 0` for some
 `a` in `S`. -/
 @[simps!]
-def torsion' (S : Type w) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S R M] :
+def torsion' (S : Type*) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S R M] :
     Submodule R M :=
   { torsion'AddSubMonoid M S with
     smul_mem' := fun a x ⟨b, h⟩ => ⟨b, by rw [smul_comm, h, smul_zero]⟩}

This is less exhaustive than its sibling #11486 because edge cases are harder to classify. No fundamental difficulty, just me being a bit fast and lazy.

Reduce the diff of #11203

@@ -865,7 +865,7 @@ theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
   refine' ⟨fun h x => _, fun h x => _⟩
   · obtain ⟨n, h0, hn⟩ := (h x).exists_nsmul_eq_zero
     exact
-      ⟨⟨n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt <| Int.coe_nat_pos.mpr h0⟩,
+      ⟨⟨n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt <| Int.natCast_pos.mpr h0⟩,
         (natCast_zsmul _ _).trans hn⟩
   · rw [isOfFinAddOrder_iff_nsmul_eq_zero]
     obtain ⟨n, hn⟩ := @h x

Purely automatic replacement. If this is in any way controversial; I'm happy to just close this PR.

@@ -68,7 +68,7 @@ section TorsionOf
 
 variable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
 
-/-- The torsion ideal of `x`, containing all `a` such that `a • x = 0`.-/
+/-- The torsion ideal of `x`, containing all `a` such that `a • x = 0`. -/
 @[simps!]
 def torsionOf (x : M) : Ideal R :=
   -- Porting note (#11036): broken dot notation on LinearMap.ker Lean4#1910
@@ -129,7 +129,7 @@ section
 
 variable (R M : Type*) [Ring R] [AddCommGroup M] [Module R M]
 
-/-- The span of `x` in `M` is isomorphic to `R` quotiented by the torsion ideal of `x`.-/
+/-- The span of `x` in `M` is isomorphic to `R` quotiented by the torsion ideal of `x`. -/
 noncomputable def quotTorsionOfEquivSpanSingleton (x : M) : (R ⧸ torsionOf R M x) ≃ₗ[R] R ∙ x :=
   (LinearMap.toSpanSingleton R M x).quotKerEquivRange.trans <|
     LinearEquiv.ofEq _ _ (LinearMap.span_singleton_eq_range R M x).symm
@@ -174,7 +174,7 @@ set_option autoImplicit true in
 -- Porting note: torsion' had metavariables and factoring out this fixed it
 -- perhaps there is a better fix
 /-- The additive submonoid of all elements `x` of `M` such that `a • x = 0`
-for some `a` in `S`.-/
+for some `a` in `S`. -/
 @[simps!]
 def torsion'AddSubMonoid (S : Type w) [CommMonoid S] [DistribMulAction S M] :
     AddSubmonoid M where
@@ -480,7 +480,7 @@ open BigOperators
 variable {ι : Type*} [DecidableEq ι] {S : Finset ι}
 
 /-- If the `p i` are pairwise coprime, a `⨅ i, p i`-torsion module is the internal direct sum of
-its `p i`-torsion submodules.-/
+its `p i`-torsion submodules. -/
 theorem torsionBySet_isInternal {p : ι → Ideal R}
     (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
     (hM : Module.IsTorsionBySet R M (⨅ i ∈ S, p i : Ideal R)) :
@@ -495,7 +495,7 @@ theorem torsionBySet_isInternal {p : ι → Ideal R}
 #align submodule.torsion_by_set_is_internal Submodule.torsionBySet_isInternal
 
 /-- If the `q i` are pairwise coprime, a `∏ i, q i`-torsion module is the internal direct sum of
-its `q i`-torsion submodules.-/
+its `q i`-torsion submodules. -/
 theorem torsionBy_isInternal {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCoprime on q))
     (hM : Module.IsTorsionBy R M <| ∏ i in S, q i) :
     DirectSum.IsInternal fun i : S => torsionBy R M <| q i := by
@@ -790,7 +790,7 @@ theorem isTorsion'_powers_iff (p : R) :
 #align submodule.is_torsion'_powers_iff Submodule.isTorsion'_powers_iff
 
 /-- In a `p ^ ∞`-torsion module (that is, a module where all elements are cancelled by scalar
-multiplication by some power of `p`), the smallest `n` such that `p ^ n • x = 0`.-/
+multiplication by some power of `p`), the smallest `n` such that `p ^ n • x = 0`. -/
 def pOrder {p : R} (hM : IsTorsion' M <| Submonoid.powers p) (x : M)
     [∀ n : ℕ, Decidable (p ^ n • x = 0)] :=
   Nat.find <| (isTorsion'_powers_iff p).mp hM x

... and add a deprecated alias for the old name. This is mostly just me discovering the power of F2

@@ -866,7 +866,7 @@ theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
   · obtain ⟨n, h0, hn⟩ := (h x).exists_nsmul_eq_zero
     exact
       ⟨⟨n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt <| Int.coe_nat_pos.mpr h0⟩,
-        (coe_nat_zsmul _ _).trans hn⟩
+        (natCast_zsmul _ _).trans hn⟩
   · rw [isOfFinAddOrder_iff_nsmul_eq_zero]
     obtain ⟨n, hn⟩ := @h x
     exact ⟨_, Int.natAbs_pos.2 (nonZeroDivisors.coe_ne_zero n), natAbs_nsmul_eq_zero.2 hn⟩

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)

@@ -391,7 +391,6 @@ section Coprime
 open BigOperators
 
 variable {ι : Type*} {p : ι → Ideal R} {S : Finset ι}
-
 variable (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
 
 -- Porting note: mem_iSup_finset_iff_exists_sum now requires DecidableEq ι
@@ -621,7 +620,6 @@ section Torsion'
 open Module
 
 variable [CommSemiring R] [AddCommMonoid M] [Module R M]
-
 variable (S : Type*) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S R M]
 
 @[simp]

... or reduce its scope (the full removal is not as obvious).

@@ -62,9 +62,6 @@ import Mathlib.Data.Set.Lattice
 Torsion, submodule, module, quotient
 -/
 
-set_option autoImplicit true
-
-
 namespace Ideal
 
 section TorsionOf
@@ -173,6 +170,7 @@ def torsionBySet (s : Set R) : Submodule R M :=
   sInf (torsionBy R M '' s)
 #align submodule.torsion_by_set Submodule.torsionBySet
 
+set_option autoImplicit true in
 -- Porting note: torsion' had metavariables and factoring out this fixed it
 -- perhaps there is a better fix
 /-- The additive submonoid of all elements `x` of `M` such that `a • x = 0`
@@ -187,6 +185,7 @@ def torsion'AddSubMonoid (S : Type w) [CommMonoid S] [DistribMulAction S M] :
     rw [smul_add, mul_smul, mul_comm, mul_smul, hx, hy, smul_zero, smul_zero, add_zero]
   zero_mem' := ⟨1, smul_zero 1⟩
 
+set_option autoImplicit true in
 /-- The `S`-torsion submodule, containing all elements `x` of `M` such that `a • x = 0` for some
 `a` in `S`. -/
 @[simps!]
@@ -833,6 +832,7 @@ namespace Ideal.Quotient
 
 open Submodule
 
+universe w
 theorem torsionBy_eq_span_singleton {R : Type w} [CommRing R] (a b : R) (ha : a ∈ R⁰) :
     torsionBy R (R ⧸ R ∙ a * b) a = R ∙ mk (R ∙ a * b) b := by
   ext x; rw [mem_torsionBy_iff, Submodule.mem_span_singleton]

• Prove isSemisimple_of_mem_adjoin: if two commuting endomorphisms of a finite-dimensional vector space over a perfect field are both semisimple, then every endomorphism in the algebra generated by them (in particular their product and sum) is semisimple.

• In the same file LinearAlgebra/Semisimple.lean, eq_zero_of_isNilpotent_isSemisimple and isSemisimple_of_squarefree_aeval_eq_zero are golfed, and IsSemisimple.minpoly_squarefree is proved

RingTheory/SimpleModule.lean:

• Define IsSemisimpleRing R to mean that R is a semisimple R-module. add properties of simple modules and a characterization (they are exactly the quotients of the ring by maximal left ideals).

• The annihilator of a semisimple module is a radical ideal.

• Any module over a semisimple ring is semisimple.

• A finite product of semisimple rings is semisimple.

• Any quotient of a semisimple ring is semisimple.

• Add Artin--Wedderburn as a TODO (proof_wanted).

• Order/Atoms.lean: add the instance from IsSimpleOrder to ComplementedLattice, so that IsSimpleModule → IsSemisimpleModule is automatically inferred.

Prerequisites for showing a product of semisimple rings is semisimple:

• Algebra/Module/Submodule/Map.lean: generalize orderIsoMapComap so that it only requires RingHomSurjective rather than RingHomInvPair

• Algebra/Ring/CompTypeclasses.lean, Mathlib/Algebra/Ring/Pi.lean, Algebra/Ring/Prod.lean: add RingHomSurjective instances

RingTheory/Artinian.lean:

• quotNilradicalEquivPi: the quotient of a commutative Artinian ring R by its nilradical is isomorphic to the (finite) product of its quotients by maximal ideals (therefore a product of fields). equivPi: if the ring is moreover reduced, then the ring itself is a product of fields. Deduce that R is a semisimple ring and both R and R[X] are decomposition monoids. Requires RingEquiv.quotientBot in RingTheory/Ideal/QuotientOperations.lean.

• Data/Polynomial/Eval.lean: the polynomial ring over a finite product of rings is isomorphic to the product of polynomial rings over individual rings. (Used to show R[X] is a decomposition monoid.)

Other necessary results:

• FieldTheory/Minpoly/Field.lean: the minimal polynomial of an element in a reduced algebra over a field is radical.

• RingTheory/PowerBasis.lean: generalize PowerBasis.finiteDimensional and rename it to .finite.

Annihilator stuff, some of which do not end up being used:

• RingTheory/Ideal/Operations.lean: define Module.annihilator and redefine Submodule.annihilator in terms of it; add lemmas, including one that says an arbitrary intersection of radical ideals is radical. The new lemma Ideal.isRadical_iff_pow_one_lt depends on pow_imp_self_of_one_lt in Mathlib/Data/Nat/Interval.lean, which is also used to golf the proof of isRadical_iff_pow_one_lt.

• Algebra/Module/Torsion.lean: add a lemma and an instance (unused)

• Data/Polynomial/Module/Basic.lean: add a def (unused) and a lemma

• LinearAlgebra/AnnihilatingPolynomial.lean: add lemma span_minpoly_eq_annihilator

Some results about idempotent linear maps (projections) and idempotent elements, used to show that any (left) ideal in a semisimple ring is spanned by an idempotent element (unused):

• LinearAlgebra/Projection.lean: add def isIdempotentElemEquiv

• LinearAlgebra/Span.lean: add two lemmas

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

@@ -232,6 +232,9 @@ def IsTorsion :=
   ∀ ⦃x : M⦄, ∃ a : R⁰, a • x = 0
 #align module.is_torsion Module.IsTorsion
 
+theorem isTorsionBySet_annihilator : IsTorsionBySet R M (Module.annihilator R M) :=
+  fun _ r ↦ Module.mem_annihilator.mp r.2 _
+
 end Module
 
 end Defs
@@ -542,6 +545,11 @@ instance IsTorsionBySet.isScalarTower
     (fun b d x => Quotient.inductionOn' d fun c => (smul_assoc b c x : _))
 #align module.is_torsion_by_set.is_scalar_tower Module.IsTorsionBySet.isScalarTower
 
+/-- Any module is also a modle over the quotient of the ring by the annihilator.
+Not an instance because it causes synthesis failures / timeouts. -/
+def quotientAnnihilator : Module (R ⧸ Module.annihilator R M) M :=
+  (isTorsionBySet_annihilator R M).module
+
 instance : Module (R ⧸ I) (M ⧸ I • (⊤ : Submodule R M)) :=
   IsTorsionBySet.module (R := R) (I := I) fun x r => by
     induction x using Quotient.inductionOn

and some subgroup results.

From PFR

@@ -738,13 +738,6 @@ lemma torsion_int {G} [AddCommGroup G] :
   refine ((isOfFinAddOrder_iff_zsmul_eq_zero (x := x)).trans ?_).symm
   simp [mem_nonZeroDivisors_iff_ne_zero]
 
-lemma AddMonoid.IsTorsionFree_iff_noZeroSMulDivisors {G : Type*} [AddCommGroup G] :
-    AddMonoid.IsTorsionFree G ↔ NoZeroSMulDivisors ℤ G := by
-  rw [Submodule.noZeroSMulDivisors_iff_torsion_eq_bot,
-    AddMonoid.isTorsionFree_iff_torsion_eq_bot,
-    ← Submodule.toAddSubgroup_injective.eq_iff,
-    Submodule.torsion_int, Submodule.bot_toAddSubgroup]
-
 end Torsion
 
 namespace QuotientTorsion

Classifies by adding issue number #11036 to porting notes claiming:

broken dot notation

@@ -74,7 +74,7 @@ variable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
 /-- The torsion ideal of `x`, containing all `a` such that `a • x = 0`.-/
 @[simps!]
 def torsionOf (x : M) : Ideal R :=
-  -- Porting note: broken dot notation on LinearMap.ker Lean4#1910
+  -- Porting note (#11036): broken dot notation on LinearMap.ker Lean4#1910
   LinearMap.ker (LinearMap.toSpanSingleton R M x)
 #align ideal.torsion_of Ideal.torsionOf
 
@@ -163,7 +163,7 @@ namespace Submodule
   `a • x = 0`. -/
 @[simps!]
 def torsionBy (a : R) : Submodule R M :=
-  -- Porting note: broken dot notation on LinearMap.ker Lean4#1910
+  -- Porting note (#11036): broken dot notation on LinearMap.ker Lean4#1910
   LinearMap.ker (DistribMulAction.toLinearMap R M a)
 #align submodule.torsion_by Submodule.torsionBy
 

I replaced a few "terminal" refine/refine's with exact.

The strategy was very simple-minded: essentially any refine whose following line had smaller indentation got replaced by exact and then I cleaned up the mess.

This PR certainly leaves some further terminal refines, but maybe the current change is beneficial.

@@ -858,7 +858,7 @@ theorem isTorsion_iff_isTorsion_nat [AddCommMonoid M] :
     exact ⟨⟨n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt h0⟩, hn⟩
   · rw [isOfFinAddOrder_iff_nsmul_eq_zero]
     obtain ⟨n, hn⟩ := @h x
-    refine' ⟨n, Nat.pos_of_ne_zero (nonZeroDivisors.coe_ne_zero _), hn⟩
+    exact ⟨n, Nat.pos_of_ne_zero (nonZeroDivisors.coe_ne_zero _), hn⟩
 #align add_monoid.is_torsion_iff_is_torsion_nat AddMonoid.isTorsion_iff_isTorsion_nat
 
 theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :

Classify by adding issue number (#10618) to porting notes claiming anything semantically equivalent to simp can prove this or simp can simplify this.

@@ -622,7 +622,7 @@ theorem mem_torsion'_iff (x : M) : x ∈ torsion' R M S ↔ ∃ a : S, a • x =
   Iff.rfl
 #align submodule.mem_torsion'_iff Submodule.mem_torsion'_iff
 
--- @[simp] Porting note : simp can prove this
+-- @[simp] Porting note (#10618): simp can prove this
 theorem mem_torsion_iff (x : M) : x ∈ torsion R M ↔ ∃ a : R⁰, a • x = 0 :=
   Iff.rfl
 #align submodule.mem_torsion_iff Submodule.mem_torsion_iff
@@ -661,7 +661,7 @@ theorem torsion'_torsion'_eq_top : torsion' R (torsion' R M S) S = ⊤ :=
 
 /-- The torsion submodule of the torsion submodule (viewed as a module) is the full
 torsion module. -/
--- @[simp] Porting note: simp can prove this
+-- @[simp] Porting note (#10618): simp can prove this
 theorem torsion_torsion_eq_top : torsion R (torsion R M) = ⊤ :=
   torsion'_torsion'_eq_top R⁰
 #align submodule.torsion_torsion_eq_top Submodule.torsion_torsion_eq_top

No changes to tactic file, it's just boring fixes throughout the library.

This follows on from #6964.

Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -415,8 +415,7 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
     · rw [mem_torsionBySet_iff] at hx ⊢
       rintro ⟨a, ha⟩
       rw [smul_smul]
-      suffices : a * μ i ∈ ⨅ i ∈ S, p i
-      exact hx ⟨_, this⟩
+      suffices a * μ i ∈ ⨅ i ∈ S, p i from hx ⟨_, this⟩
       rw [mem_iInf]
       intro j
       rw [mem_iInf]

and replace it by two more explicit lemmas

@@ -871,7 +871,7 @@ theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
         (coe_nat_zsmul _ _).trans hn⟩
   · rw [isOfFinAddOrder_iff_nsmul_eq_zero]
     obtain ⟨n, hn⟩ := @h x
-    exact exists_nsmul_eq_zero_of_zsmul_eq_zero (nonZeroDivisors.coe_ne_zero n) hn
+    exact ⟨_, Int.natAbs_pos.2 (nonZeroDivisors.coe_ne_zero n), natAbs_nsmul_eq_zero.2 hn⟩
 #align add_monoid.is_torsion_iff_is_torsion_int AddMonoid.isTorsion_iff_isTorsion_int
 
 end AddMonoid

@@ -9,6 +9,7 @@ import Mathlib.LinearAlgebra.Isomorphisms
 import Mathlib.GroupTheory.Torsion
 import Mathlib.RingTheory.Coprime.Ideal
 import Mathlib.RingTheory.Finiteness
+import Mathlib.Data.Set.Lattice
 
 #align_import algebra.module.torsion from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
 

@@ -392,7 +392,7 @@ variable {ι : Type*} {p : ι → Ideal R} {S : Finset ι}
 variable (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
 
 -- Porting note: mem_iSup_finset_iff_exists_sum now requires DecidableEq ι
-theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf [DecidableEq ι] :
+theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf :
     ⨆ i ∈ S, torsionBySet R M (p i) = torsionBySet R M ↑(⨅ i ∈ S, p i) := by
   rcases S.eq_empty_or_nonempty with h | h
   · simp only [h]
@@ -430,7 +430,7 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf [DecidableEq ι] :
 #align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf
 
 -- Porting note: iSup_torsionBySet_ideal_eq_torsionBySet_iInf now requires DecidableEq ι
-theorem supIndep_torsionBySet_ideal [DecidableEq ι] : S.SupIndep fun i => torsionBySet R M <| p i :=
+theorem supIndep_torsionBySet_ideal : S.SupIndep fun i => torsionBySet R M <| p i :=
   fun T hT i hi hiT => by
   rw [disjoint_iff, Finset.sup_eq_iSup,
     iSup_torsionBySet_ideal_eq_torsionBySet_iInf fun i hi j hj ij => hp (hT hi) (hT hj) ij]
@@ -443,7 +443,7 @@ theorem supIndep_torsionBySet_ideal [DecidableEq ι] : S.SupIndep fun i => torsi
 
 variable {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCoprime on q))
 
-theorem iSup_torsionBy_eq_torsionBy_prod [DecidableEq ι] :
+theorem iSup_torsionBy_eq_torsionBy_prod :
     ⨆ i ∈ S, torsionBy R M (q i) = torsionBy R M (∏ i in S, q i) := by
   rw [← torsionBySet_span_singleton_eq, Ideal.submodule_span_eq, ←
     Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf, ←
@@ -456,8 +456,7 @@ theorem iSup_torsionBy_eq_torsionBy_prod [DecidableEq ι] :
   exact fun i hi j hj ij => (Ideal.sup_eq_top_iff_isCoprime _ _).mpr (hq hi hj ij)
 #align submodule.supr_torsion_by_eq_torsion_by_prod Submodule.iSup_torsionBy_eq_torsionBy_prod
 
--- Porting note: supIndep_torsionBySet_ideal now requires DecidableEq ι
-theorem supIndep_torsionBy [DecidableEq ι] : S.SupIndep fun i => torsionBy R M <| q i := by
+theorem supIndep_torsionBy : S.SupIndep fun i => torsionBy R M <| q i := by
   convert supIndep_torsionBySet_ideal (M := M) fun i hi j hj ij =>
       (Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij
   exact (torsionBySet_span_singleton_eq (R := R) (M := M) _).symm

(provided the coefficients are a principal ideal ring)

@@ -568,7 +568,7 @@ instance (I : Ideal R) {S : Type*} [SMul S R] [SMul S M] [IsScalarTower S R M]
   inferInstance
 
 /-- The `a`-torsion submodule as an `(R ⧸ R∙a)`-module. -/
-instance (a : R) : Module (R ⧸ R ∙ a) (torsionBy R M a) :=
+instance instModuleQuotientTorsionBy (a : R) : Module (R ⧸ R ∙ a) (torsionBy R M a) :=
   Module.IsTorsionBySet.module <|
     (Module.isTorsionBySet_span_singleton_iff a).mpr <| torsionBy_isTorsionBy a
 
@@ -591,6 +591,19 @@ instance (a : R) {S : Type*} [SMul S R] [SMul S M] [IsScalarTower S R M] [IsScal
     IsScalarTower S (R ⧸ R ∙ a) (torsionBy R M a) :=
   inferInstance
 
+/-- Given an `R`-module `M` and an element `a` in `R`, submodules of the `a`-torsion submodule of
+`M` do not depend on whether we take scalars to be `R` or `R ⧸ R ∙ a`. -/
+def submodule_torsionBy_orderIso (a : R) :
+    Submodule (R ⧸ R ∙ a) (torsionBy R M a) ≃o Submodule R (torsionBy R M a) :=
+  { restrictScalarsEmbedding R (R ⧸ R ∙ a) (torsionBy R M a) with
+    invFun := fun p ↦
+      { carrier := p
+        add_mem' := add_mem
+        zero_mem' := p.zero_mem
+        smul_mem' := by rintro ⟨b⟩; exact p.smul_mem b }
+    left_inv := by intro; ext; simp [restrictScalarsEmbedding]
+    right_inv := by intro; ext; simp [restrictScalarsEmbedding] }
+
 end Submodule
 
 end NeedsGroup

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

@@ -720,6 +720,19 @@ theorem noZeroSMulDivisors_iff_torsion_eq_bot : NoZeroSMulDivisors R M ↔ torsi
             exact ⟨⟨a, mem_nonZeroDivisors_of_ne_zero ha⟩, hax⟩ }
 #align submodule.no_zero_smul_divisors_iff_torsion_eq_bot Submodule.noZeroSMulDivisors_iff_torsion_eq_bot
 
+lemma torsion_int {G} [AddCommGroup G] :
+    (torsion ℤ G).toAddSubgroup = AddCommGroup.torsion G := by
+  ext x
+  refine ((isOfFinAddOrder_iff_zsmul_eq_zero (x := x)).trans ?_).symm
+  simp [mem_nonZeroDivisors_iff_ne_zero]
+
+lemma AddMonoid.IsTorsionFree_iff_noZeroSMulDivisors {G : Type*} [AddCommGroup G] :
+    AddMonoid.IsTorsionFree G ↔ NoZeroSMulDivisors ℤ G := by
+  rw [Submodule.noZeroSMulDivisors_iff_torsion_eq_bot,
+    AddMonoid.isTorsionFree_iff_torsion_eq_bot,
+    ← Submodule.toAddSubgroup_injective.eq_iff,
+    Submodule.torsion_int, Submodule.bot_toAddSubgroup]
+
 end Torsion
 
 namespace QuotientTorsion

I literally went through and regex'd some uses of cases', replacing them with rcases; this is meant to be a low effort PR as I hope that tools can do this in the future.

rcases is an easier replacement than cases, though with better tools we could in future do a second pass converting simple rcases added here (and existing ones) to cases.

@@ -394,7 +394,7 @@ variable (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
 -- Porting note: mem_iSup_finset_iff_exists_sum now requires DecidableEq ι
 theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf [DecidableEq ι] :
     ⨆ i ∈ S, torsionBySet R M (p i) = torsionBySet R M ↑(⨅ i ∈ S, p i) := by
-  cases' S.eq_empty_or_nonempty with h h
+  rcases S.eq_empty_or_nonempty with h | h
   · simp only [h]
     -- Porting note: converts were not cooperating
     convert iSup_emptyset (f := fun i => torsionBySet R M (p i)) <;> simp

This eliminates (fun a ↦ β) α in the type when applying a FunLike.

Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -375,7 +375,7 @@ variable (R M)
 
 theorem torsion_gc :
     @GaloisConnection (Submodule R M) (Ideal R)ᵒᵈ _ _ annihilator fun I =>
-      torsionBySet R M <| OrderDual.ofDual I :=
+      torsionBySet R M ↑(OrderDual.ofDual I) :=
   fun _ _ =>
   ⟨fun h x hx => (mem_torsionBySet_iff _ _).mpr fun ⟨_, ha⟩ => mem_annihilator.mp (h ha) x hx,
     fun h a ha => mem_annihilator.mpr fun _ hx => (mem_torsionBySet_iff _ _).mp (h hx) ⟨a, ha⟩⟩

Many lemmas in GroupTheory.OrderOfElement were stated for elements of finite groups even though they work more generally for torsion elements of possibly infinite groups. This PR generalises those lemmas (and leaves convenience lemmas stated for finite groups), and fixes a bunch of names to use dot notation.

Renames

• Function.eq_of_lt_minimalPeriod_of_iterate_eq → Function.iterate_injOn_Iio_minimalPeriod
• Function.eq_iff_lt_minimalPeriod_of_iterate_eq → Function.iterate_eq_iterate_iff_of_lt_minimalPeriod
• isOfFinOrder_iff_coe → Submonoid.isOfFinOrder_coe
• orderOf_pos' → IsOfFinOrder.orderOf_pos
• pow_eq_mod_orderOf → pow_mod_orderOf (and turned around)
• pow_injective_of_lt_orderOf → pow_injOn_Iio_orderOf
• mem_powers_iff_mem_range_order_of' → IsOfFinOrder.mem_powers_iff_mem_range_orderOf
• orderOf_pow'' → IsOfFinOrder.orderOf_pow
• orderOf_pow_coprime → Nat.Coprime.orderOf_pow
• zpow_eq_mod_orderOf → zpow_mod_orderOf (and turned around)
• exists_pow_eq_one → isOfFinOrder_of_finite
• pow_apply_eq_pow_mod_orderOf_cycleOf_apply → pow_mod_orderOf_cycleOf_apply

New lemmas

• IsOfFinOrder.powers_eq_image_range_orderOf
• IsOfFinOrder.natCard_powers_le_orderOf
• IsOfFinOrder.finite_powers
• finite_powers
• infinite_powers
• Nat.card_submonoidPowers
• IsOfFinOrder.mem_powers_iff_mem_zpowers
• IsOfFinOrder.powers_eq_zpowers
• IsOfFinOrder.mem_zpowers_iff_mem_range_orderOf
• IsOfFinOrder.exists_pow_eq_one

Other changes

• Move decidableMemPowers/fintypePowers to GroupTheory.Submonoid.Membership and decidableMemZpowers/fintypeZpowers to GroupTheory.Subgroup.ZPowers.
• finEquivPowers, finEquivZpowers, powersEquivPowers and zpowersEquivZpowers now assume IsOfFinTorsion x instead of Finite G.
• isOfFinOrder_iff_pow_eq_one now takes one less explicit argument.
• Delete Equiv.Perm.IsCycle.exists_pow_eq_one since it was saying that a permutation over a finite type is torsion, but this is trivial since the group of permutation is itself finite, so we can use isOfFinOrder_of_finite instead.

@@ -829,7 +829,7 @@ namespace AddMonoid
 theorem isTorsion_iff_isTorsion_nat [AddCommMonoid M] :
     AddMonoid.IsTorsion M ↔ Module.IsTorsion ℕ M := by
   refine' ⟨fun h x => _, fun h x => _⟩
-  · obtain ⟨n, h0, hn⟩ := (isOfFinAddOrder_iff_nsmul_eq_zero x).mp (h x)
+  · obtain ⟨n, h0, hn⟩ := (h x).exists_nsmul_eq_zero
     exact ⟨⟨n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt h0⟩, hn⟩
   · rw [isOfFinAddOrder_iff_nsmul_eq_zero]
     obtain ⟨n, hn⟩ := @h x
@@ -839,7 +839,7 @@ theorem isTorsion_iff_isTorsion_nat [AddCommMonoid M] :
 theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
     AddMonoid.IsTorsion M ↔ Module.IsTorsion ℤ M := by
   refine' ⟨fun h x => _, fun h x => _⟩
-  · obtain ⟨n, h0, hn⟩ := (isOfFinAddOrder_iff_nsmul_eq_zero x).mp (h x)
+  · obtain ⟨n, h0, hn⟩ := (h x).exists_nsmul_eq_zero
     exact
       ⟨⟨n, mem_nonZeroDivisors_of_ne_zero <| ne_of_gt <| Int.coe_nat_pos.mpr h0⟩,
         (coe_nat_zsmul _ _).trans hn⟩

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -426,8 +426,7 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf [DecidableEq ι] :
       · have := coe_mem (μ i)
         simp only [mem_iInf] at this
         exact Ideal.mul_mem_left _ _ (this j hj ij)
-    · simp_rw [coe_mk]
-      rw [← Finset.sum_smul, hμ, one_smul]
+    · rw [← Finset.sum_smul, hμ, one_smul]
 #align submodule.supr_torsion_by_ideal_eq_torsion_by_infi Submodule.iSup_torsionBySet_ideal_eq_torsionBySet_iInf
 
 -- Porting note: iSup_torsionBySet_ideal_eq_torsionBySet_iInf now requires DecidableEq ι

Autoimplicits are highly controversial and also defeat the performance-improving work in #6474.

The intent of this PR is to make autoImplicit opt-in on a per-file basis, by disabling it in the lakefile and enabling it again with set_option autoImplicit true in the few files that rely on it.

That also keeps this PR small, as opposed to attempting to "fix" files to not need it any more.

I claim that many of the uses of autoImplicit in these files are accidental; situations such as:

• Assuming variables are in scope, but pasting the lemma in the wrong section
• Pasting in a lemma from a scratch file without checking to see if the variable names are consistent with the rest of the file
• Making a copy-paste error between lemmas and forgetting to add an explicit arguments.

Having set_option autoImplicit false as the default prevents these types of mistake being made in the 90% of files where autoImplicits are not used at all, and causes them to be caught by CI during review.

I think there were various points during the port where we encouraged porters to delete the universes u v lines; I think having autoparams for universe variables only would cover a lot of the cases we actually use them, while avoiding any real shortcomings.

A Zulip poll (after combining overlapping votes accordingly) was in favor of this change with 5:5:18 as the no:dontcare:yes vote ratio.

While this PR was being reviewed, a handful of files gained some more likely-accidental autoImplicits. In these places, set_option autoImplicit true has been placed locally within a section, rather than at the top of the file.

@@ -61,6 +61,8 @@ import Mathlib.RingTheory.Finiteness
 Torsion, submodule, module, quotient
 -/
 
+set_option autoImplicit true
+
 
 namespace Ideal
 

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

This has nice performance benefits.

@@ -66,7 +66,7 @@ namespace Ideal
 
 section TorsionOf
 
-variable (R M : Type _) [Semiring R] [AddCommMonoid M] [Module R M]
+variable (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
 
 /-- The torsion ideal of `x`, containing all `a` such that `a • x = 0`.-/
 @[simps!]
@@ -107,7 +107,7 @@ theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDiv
 
 /-- See also `CompleteLattice.Independent.linearIndependent` which provides the same conclusion
 but requires the stronger hypothesis `NoZeroSMulDivisors R M`. -/
-theorem CompleteLattice.Independent.linear_independent' {ι R M : Type _} {v : ι → M} [Ring R]
+theorem CompleteLattice.Independent.linear_independent' {ι R M : Type*} {v : ι → M} [Ring R]
     [AddCommGroup M] [Module R M] (hv : CompleteLattice.Independent fun i => R ∙ v i)
     (h_ne_zero : ∀ i, Ideal.torsionOf R M (v i) = ⊥) : LinearIndependent R v := by
   refine' linearIndependent_iff_not_smul_mem_span.mpr fun i r hi => _
@@ -127,7 +127,7 @@ end TorsionOf
 
 section
 
-variable (R M : Type _) [Ring R] [AddCommGroup M] [Module R M]
+variable (R M : Type*) [Ring R] [AddCommGroup M] [Module R M]
 
 /-- The span of `x` in `M` is isomorphic to `R` quotiented by the torsion ideal of `x`.-/
 noncomputable def quotTorsionOfEquivSpanSingleton (x : M) : (R ⧸ torsionOf R M x) ≃ₗ[R] R ∙ x :=
@@ -152,7 +152,7 @@ open nonZeroDivisors
 
 section Defs
 
-variable (R M : Type _) [CommSemiring R] [AddCommMonoid M] [Module R M]
+variable (R M : Type*) [CommSemiring R] [AddCommMonoid M] [Module R M]
 
 namespace Submodule
 
@@ -218,7 +218,7 @@ def IsTorsionBySet (s : Set R) :=
 
 /-- An `S`-torsion module is a module where every element is `a`-torsion for some `a` in `S`. -/
 @[reducible]
-def IsTorsion' (S : Type _) [SMul S M] :=
+def IsTorsion' (S : Type*) [SMul S M] :=
   ∀ ⦃x : M⦄, ∃ a : S, a • x = 0
 #align module.is_torsion' Module.IsTorsion'
 
@@ -233,7 +233,7 @@ end Module
 
 end Defs
 
-variable {R M : Type _}
+variable {R M : Type*}
 
 section
 
@@ -385,7 +385,7 @@ section Coprime
 
 open BigOperators
 
-variable {ι : Type _} {p : ι → Ideal R} {S : Finset ι}
+variable {ι : Type*} {p : ι → Ideal R} {S : Finset ι}
 
 variable (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
 
@@ -476,7 +476,7 @@ namespace Submodule
 
 open BigOperators
 
-variable {ι : Type _} [DecidableEq ι] {S : Finset ι}
+variable {ι : Type*} [DecidableEq ι] {S : Finset ι}
 
 /-- If the `p i` are pairwise coprime, a `⨅ i, p i`-torsion module is the internal direct sum of
 its `p i`-torsion submodules.-/
@@ -535,7 +535,7 @@ def IsTorsionBySet.module : Module (R ⧸ I) M :=
 #align module.is_torsion_by_set.module Module.IsTorsionBySet.module
 
 instance IsTorsionBySet.isScalarTower
-    {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M] [IsScalarTower S R R] :
+    {S : Type*} [SMul S R] [SMul S M] [IsScalarTower S R M] [IsScalarTower S R R] :
     @IsScalarTower S (R ⧸ I) M _ (IsTorsionBySet.module hM).toSMul _ :=
   -- Porting note: still needed to be fed the Module R / I M instance
   @IsScalarTower.mk S (R ⧸ I) M _ (IsTorsionBySet.module hM).toSMul _
@@ -562,7 +562,7 @@ theorem torsionBySet.mk_smul (I : Ideal R) (b : R) (x : torsionBySet R M I) :
   rfl
 #align submodule.torsion_by_set.mk_smul Submodule.torsionBySet.mk_smul
 
-instance (I : Ideal R) {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M]
+instance (I : Ideal R) {S : Type*} [SMul S R] [SMul S M] [IsScalarTower S R M]
     [IsScalarTower S R R] : IsScalarTower S (R ⧸ I) (torsionBySet R M I) :=
   inferInstance
 
@@ -586,7 +586,7 @@ theorem torsionBy.mk_smul (a b : R) (x : torsionBy R M a) :
   rfl
 #align submodule.torsion_by.mk_smul Submodule.torsionBy.mk_smul
 
-instance (a : R) {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M] [IsScalarTower S R R] :
+instance (a : R) {S : Type*} [SMul S R] [SMul S M] [IsScalarTower S R M] [IsScalarTower S R R] :
     IsScalarTower S (R ⧸ R ∙ a) (torsionBy R M a) :=
   inferInstance
 
@@ -602,7 +602,7 @@ open Module
 
 variable [CommSemiring R] [AddCommMonoid M] [Module R M]
 
-variable (S : Type _) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S R M]
+variable (S : Type*) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S R M]
 
 @[simp]
 theorem mem_torsion'_iff (x : M) : x ∈ torsion' R M S ↔ ∃ a : S, a • x = 0 :=

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Pierre-Alexandre Bazin
-
-! This file was ported from Lean 3 source module algebra.module.torsion
-! leanprover-community/mathlib commit cdc34484a07418af43daf8198beaf5c00324bca8
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.DirectSum.Module
 import Mathlib.Algebra.Module.BigOperators
@@ -15,6 +10,8 @@ import Mathlib.GroupTheory.Torsion
 import Mathlib.RingTheory.Coprime.Ideal
 import Mathlib.RingTheory.Finiteness
 
+#align_import algebra.module.torsion from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
+
 /-!
 # Torsion submodules
 

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

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

@@ -196,7 +196,7 @@ def torsion' (S : Type w) [CommMonoid S] [DistribMulAction S M] [SMulCommClass S
     smul_mem' := fun a x ⟨b, h⟩ => ⟨b, by rw [smul_comm, h, smul_zero]⟩}
 #align submodule.torsion' Submodule.torsion'
 
-/-- The torsion submodule, containing all elements `x` of `M` such that  `a • x = 0` for some
+/-- The torsion submodule, containing all elements `x` of `M` such that `a • x = 0` for some
   non-zero-divisor `a` in `R`. -/
 @[reducible]
 def torsion :=

@@ -394,7 +394,7 @@ variable (hp : (S : Set ι).Pairwise fun i j => p i ⊔ p j = ⊤)
 
 -- Porting note: mem_iSup_finset_iff_exists_sum now requires DecidableEq ι
 theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf [DecidableEq ι] :
-    (⨆ i ∈ S, torsionBySet R M <| p i) = torsionBySet R M ↑(⨅ i ∈ S, p i) := by
+    ⨆ i ∈ S, torsionBySet R M (p i) = torsionBySet R M ↑(⨅ i ∈ S, p i) := by
   cases' S.eq_empty_or_nonempty with h h
   · simp only [h]
     -- Porting note: converts were not cooperating
@@ -446,7 +446,7 @@ theorem supIndep_torsionBySet_ideal [DecidableEq ι] : S.SupIndep fun i => torsi
 variable {q : ι → R} (hq : (S : Set ι).Pairwise <| (IsCoprime on q))
 
 theorem iSup_torsionBy_eq_torsionBy_prod [DecidableEq ι] :
-    (⨆ i ∈ S, torsionBy R M <| q i) = torsionBy R M (∏ i in S, q i) := by
+    ⨆ i ∈ S, torsionBy R M (q i) = torsionBy R M (∏ i in S, q i) := by
   rw [← torsionBySet_span_singleton_eq, Ideal.submodule_span_eq, ←
     Ideal.finset_inf_span_singleton _ _ hq, Finset.inf_eq_iInf, ←
     iSup_torsionBySet_ideal_eq_torsionBySet_iInf]

Changes are of the form

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

@@ -412,7 +412,7 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf [DecidableEq ι] :
       (mem_iSup_finset_iff_exists_sum _ _).mp
         ((Ideal.eq_top_iff_one _).mp <| (Ideal.iSup_iInf_eq_top_iff_pairwise h _).mpr hp)
     refine' ⟨fun i => ⟨(μ i : R) • x, _⟩, _⟩
-    · rw [mem_torsionBySet_iff] at hx⊢
+    · rw [mem_torsionBySet_iff] at hx ⊢
       rintro ⟨a, ha⟩
       rw [smul_smul]
       suffices : a * μ i ∈ ⨅ i ∈ S, p i

All of these are doc fixes

@@ -29,9 +29,9 @@ import Mathlib.RingTheory.Finiteness
   that `a • x = 0` for some `a` in `S`.
 * `Submodule.torsion R M` : the torsion submodule, containing all elements `x` of `M` such that
   `a • x = 0` for some non-zero-divisor `a` in `R`.
-* `Module.IsTorsionBy R M a` : the property that defines a `a`-torsion module. Similarly,
+* `Module.IsTorsionBy R M a` : the property that defines an `a`-torsion module. Similarly,
   `IsTorsionBySet`, `IsTorsion'` and `IsTorsion`.
-* `Module.IsTorsionBySet.module` : Creates a `R ⧸ I`-module from a `R`-module that
+* `Module.IsTorsionBySet.module` : Creates an `R ⧸ I`-module from an `R`-module that
   `IsTorsionBySet R _ I`.
 
 ## Main statements
@@ -40,7 +40,7 @@ import Mathlib.RingTheory.Finiteness
   the quotient by its torsion ideal.
 * `torsion' R M S` and `torsion R M` are submodules.
 * `torsionBySet_eq_torsionBySet_span` : torsion by a set is torsion by the ideal generated by it.
-* `Submodule.torsionBy_is_torsionBy` : the `a`-torsion submodule is a `a`-torsion module.
+* `Submodule.torsionBy_is_torsionBy` : the `a`-torsion submodule is an `a`-torsion module.
   Similar lemmas for `torsion'` and `torsion`.
 * `Submodule.torsionBy_isInternal` : a `∏ i, p i`-torsion module is the internal direct sum of its
   `p i`-torsion submodules when the `p i` are pairwise coprime. A more general version with coprime
@@ -207,7 +207,7 @@ end Submodule
 
 namespace Module
 
-/-- A `a`-torsion module is a module where every element is `a`-torsion. -/
+/-- An `a`-torsion module is a module where every element is `a`-torsion. -/
 @[reducible]
 def IsTorsionBy (a : R) :=
   ∀ ⦃x : M⦄, a • x = 0
@@ -219,7 +219,7 @@ def IsTorsionBySet (s : Set R) :=
   ∀ ⦃x : M⦄ ⦃a : s⦄, (a : R) • x = 0
 #align module.is_torsion_by_set Module.IsTorsionBySet
 
-/-- A `S`-torsion module is a module where every element is `a`-torsion for some `a` in `S`. -/
+/-- An `S`-torsion module is a module where every element is `a`-torsion for some `a` in `S`. -/
 @[reducible]
 def IsTorsion' (S : Type _) [SMul S M] :=
   ∀ ⦃x : M⦄, ∃ a : S, a • x = 0
@@ -332,7 +332,7 @@ theorem isTorsionBySet_iff_torsionBySet_eq_top :
     trivial⟩
 #align module.is_torsion_by_set_iff_torsion_by_set_eq_top Module.isTorsionBySet_iff_torsionBySet_eq_top
 
-/-- A `a`-torsion module is a module whose `a`-torsion submodule is the full space. -/
+/-- An `a`-torsion module is a module whose `a`-torsion submodule is the full space. -/
 theorem isTorsionBy_iff_torsionBy_eq_top : IsTorsionBy R M a ↔ torsionBy R M a = ⊤ := by
   rw [← torsionBySet_singleton_eq, ← isTorsionBySet_singleton_iff,
     isTorsionBySet_iff_torsionBySet_eq_top]
@@ -358,7 +358,7 @@ theorem torsionBySet_isTorsionBySet : IsTorsionBySet R (torsionBySet R M s) s :=
   Subtype.ext <| (mem_torsionBySet_iff _ _).mp hx a
 #align submodule.torsion_by_set_is_torsion_by_set Submodule.torsionBySet_isTorsionBySet
 
-/-- The `a`-torsion submodule is a `a`-torsion module. -/
+/-- The `a`-torsion submodule is an `a`-torsion module. -/
 theorem torsionBy_isTorsionBy : IsTorsionBy R (torsionBy R M a) a := fun _ => smul_torsionBy _ _
 #align submodule.torsion_by_is_torsion_by Submodule.torsionBy_isTorsionBy
 
@@ -531,7 +531,7 @@ theorem IsTorsionBySet.mk_smul (b : R) (x : M) :
   rfl
 #align module.is_torsion_by_set.mk_smul Module.IsTorsionBySet.mk_smul
 
-/-- A `(R ⧸ I)`-module is a `R`-module which `IsTorsionBySet R M I`. -/
+/-- An `(R ⧸ I)`-module is an `R`-module which `IsTorsionBySet R M I`. -/
 def IsTorsionBySet.module : Module (R ⧸ I) M :=
   @Function.Surjective.moduleLeft _ _ _ _ _ _ _ hM.hasSMul _ Ideal.Quotient.mk_surjective
     (IsTorsionBySet.mk_smul hM)
@@ -569,7 +569,7 @@ instance (I : Ideal R) {S : Type _} [SMul S R] [SMul S M] [IsScalarTower S R M]
     [IsScalarTower S R R] : IsScalarTower S (R ⧸ I) (torsionBySet R M I) :=
   inferInstance
 
-/-- The `a`-torsion submodule as a `(R ⧸ R∙a)`-module. -/
+/-- The `a`-torsion submodule as an `(R ⧸ R∙a)`-module. -/
 instance (a : R) : Module (R ⧸ R ∙ a) (torsionBy R M a) :=
   Module.IsTorsionBySet.module <|
     (Module.isTorsionBySet_span_singleton_iff a).mpr <| torsionBy_isTorsionBy a
@@ -633,14 +633,14 @@ instance : DistribMulAction S (torsion' R M S) :=
 instance : SMulCommClass S R (torsion' R M S) :=
   ⟨fun _ _ _ => Subtype.ext <| smul_comm _ _ _⟩
 
-/-- A `S`-torsion module is a module whose `S`-torsion submodule is the full space. -/
+/-- An `S`-torsion module is a module whose `S`-torsion submodule is the full space. -/
 theorem isTorsion'_iff_torsion'_eq_top : IsTorsion' M S ↔ torsion' R M S = ⊤ :=
   ⟨fun h => eq_top_iff.mpr fun _ _ => @h _, fun h x => by
     rw [← @mem_torsion'_iff R, h]
     trivial⟩
 #align submodule.is_torsion'_iff_torsion'_eq_top Submodule.isTorsion'_iff_torsion'_eq_top
 
-/-- The `S`-torsion submodule is a `S`-torsion module. -/
+/-- The `S`-torsion submodule is an `S`-torsion module. -/
 theorem torsion'_isTorsion' : IsTorsion' (torsion' R M S) S := fun ⟨_, ⟨a, h⟩⟩ => ⟨a, Subtype.ext h⟩
 #align submodule.torsion'_is_torsion' Submodule.torsion'_isTorsion'
 

These are all doc fixes

@@ -491,7 +491,7 @@ theorem torsionBySet_isInternal {p : ι → Ideal R}
     (CompleteLattice.independent_iff_supIndep.mpr <| supIndep_torsionBySet_ideal hp)
     (by
       apply (iSup_subtype'' ↑S fun i => torsionBySet R M <| p i).trans
-      -- Porting note: timesout if we change apply below to <|
+      -- Porting note: times out if we change apply below to <|
       apply (iSup_torsionBySet_ideal_eq_torsionBySet_iInf hp).trans <|
         (Module.isTorsionBySet_iff_torsionBySet_eq_top _).mp hM)
 #align submodule.torsion_by_set_is_internal Submodule.torsionBySet_isInternal

@@ -405,7 +405,7 @@ theorem iSup_torsionBySet_ideal_eq_torsionBySet_iInf [DecidableEq ι] :
     apply iSup_le _
     intro is
     apply torsionBySet_le_torsionBySet_of_subset
-    exact (iInf_le (fun i => ⨅ _H : i ∈ S, p i) i).trans (iInf_le _ is)
+    exact (iInf_le (fun i => ⨅ _ : i ∈ S, p i) i).trans (iInf_le _ is)
   · intro x hx
     rw [mem_iSup_finset_iff_exists_sum]
     obtain ⟨μ, hμ⟩ :=

I ran codespell Mathlib and got tired halfway through the suggestions.

@@ -27,7 +27,7 @@ import Mathlib.RingTheory.Finiteness
   `a • x = 0` for all `a` in `s`.
 * `Submodule.torsion' R M S` : the `S`-torsion submodule, containing all elements `x` of `M` such
   that `a • x = 0` for some `a` in `S`.
-* `Submodule.torsion R M` : the torsion submoule, containing all elements `x` of `M` such that
+* `Submodule.torsion R M` : the torsion submodule, containing all elements `x` of `M` such that
   `a • x = 0` for some non-zero-divisor `a` in `R`.
 * `Module.IsTorsionBy R M a` : the property that defines a `a`-torsion module. Similarly,
   `IsTorsionBySet`, `IsTorsion'` and `IsTorsion`.

@@ -30,9 +30,9 @@ import Mathlib.RingTheory.Finiteness
 * `Submodule.torsion R M` : the torsion submoule, containing all elements `x` of `M` such that
   `a • x = 0` for some non-zero-divisor `a` in `R`.
 * `Module.IsTorsionBy R M a` : the property that defines a `a`-torsion module. Similarly,
-  `is_torsion_by_set`, `is_torsion'` and `is_torsion`.
+  `IsTorsionBySet`, `IsTorsion'` and `IsTorsion`.
 * `Module.IsTorsionBySet.module` : Creates a `R ⧸ I`-module from a `R`-module that
-  `is_torsion_by_set R _ I`.
+  `IsTorsionBySet R _ I`.
 
 ## Main statements
 
@@ -46,7 +46,7 @@ import Mathlib.RingTheory.Finiteness
   `p i`-torsion submodules when the `p i` are pairwise coprime. A more general version with coprime
   ideals is `Submodule.torsionBySet_is_internal`.
 * `Submodule.noZeroSMulDivisors_iff_torsion_bot` : a module over a domain has
-  `no_zero_smul_divisors` (that is, there is no non-zero `a`, `x` such that `a • x = 0`)
+  `NoZeroSMulDivisors` (that is, there is no non-zero `a`, `x` such that `a • x = 0`)
   iff its torsion submodule is trivial.
 * `Submodule.QuotientTorsion.torsion_eq_bot` : quotienting by the torsion submodule makes the
   torsion submodule of the new module trivial. If `R` is a domain, we can derive an instance
@@ -54,7 +54,7 @@ import Mathlib.RingTheory.Finiteness
 
 ## Notation
 
-* The notions are defined for a `comm_semiring R` and a `module R M`. Some additional hypotheses on
+* The notions are defined for a `CommSemiring R` and a `Module R M`. Some additional hypotheses on
   `R` and `M` are required by some lemmas.
 * The letters `a`, `b`, ... are used for scalars (in `R`), while `x`, `y`, ... are used for vectors
   (in `M`).
@@ -851,4 +851,3 @@ theorem isTorsion_iff_isTorsion_int [AddCommGroup M] :
 #align add_monoid.is_torsion_iff_is_torsion_int AddMonoid.isTorsion_iff_isTorsion_int
 
 end AddMonoid
-

Co-authored-by: Johan Commelin <johan@commelin.net>