ring_theory.principal_ideal_domain ⟷ Mathlib.RingTheory.PrincipalIdealDomain

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

@@ -201,7 +201,7 @@ theorem to_maximal_ideal [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] {S :
         exact (hxS <| hTS hxT).elim
       cases' (mem_iff_generator_dvd _).1 h with y hy
       have : generator S ≠ 0 := mt (eq_bot_iff_generator_eq_zero _).2 hS
-      rw [← mul_one (generator S), hy, mul_left_comm, mul_right_inj' this] at hz 
+      rw [← mul_one (generator S), hy, mul_left_comm, mul_right_inj' this] at hz
       exact hz.symm ▸ T.mul_mem_right _ (generator_mem T)⟩
 #align is_prime.to_maximal_ideal IsPrime.to_maximal_ideal
 -/
@@ -243,7 +243,7 @@ instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincip
                       fun h₁ => WellFounded.not_lt_min wf _ h h₁ (mod_lt x hmin.2)
                     have : x % WellFounded.min wf {x : R | x ∈ S ∧ x ≠ 0} h = 0 :=
                       by
-                      simp only [not_and_or, Set.mem_setOf_eq, not_ne_iff] at this 
+                      simp only [not_and_or, Set.mem_setOf_eq, not_ne_iff] at this
                       cases this; cases this ((mod_mem_iff hmin.1).2 hx); exact this
                     simp [*]),
               fun hx =>
@@ -421,7 +421,7 @@ variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [GCDMonoid R]
 theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
   by
   obtain ⟨d, hd⟩ := IsPrincipalIdealRing.principal (span ({x, y} : Set R))
-  rw [submodule_span_eq] at hd 
+  rw [submodule_span_eq] at hd
   rw [hd]
   suffices Associated d (gcd x y) by
     obtain ⟨D, HD⟩ := this
@@ -598,7 +598,7 @@ theorem nonPrincipals_zorn (c : Set (Ideal R)) (hs : c ⊆ nonPrincipals R)
   obtain ⟨J, hJc, hxJ⟩ := (Submodule.mem_sSup_of_directed ⟨K, hKmem⟩ hchain.directed_on).1 hxmem
   have hSupJ : Sup c = J := le_antisymm (by simp [hx, Ideal.span_le, hxJ]) (le_sSup hJc)
   specialize hs hJc
-  rw [← hSupJ, hx, nonPrincipals_def] at hs 
+  rw [← hSupJ, hx, nonPrincipals_def] at hs
   exact hs ⟨⟨x, rfl⟩⟩
 #align non_principals_zorn nonPrincipals_zorn
 -/
@@ -640,14 +640,14 @@ theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPr
       ⟨⟨a * b, (le_antisymm fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _⟩⟩
   · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
     have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
-    erw [ha, mem_span_singleton'] at hisup this 
+    erw [ha, mem_span_singleton'] at hisup this
     obtain ⟨v, rfl⟩ := this
     obtain ⟨u, rfl⟩ := hisup
     have hucolon : u ∈ I.colon (span {v * a}) :=
       by
       rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
       exact mul_mem_right _ _ hi
-    erw [hb, mem_span_singleton'] at hucolon 
+    erw [hb, mem_span_singleton'] at hucolon
     obtain ⟨z, rfl⟩ := hucolon
     exact mem_span_singleton'.2 ⟨z, by ring⟩
   · rw [← Ideal.submodule_span_eq, ← ha, Ideal.sup_mul, sup_le_iff,

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

@@ -300,17 +300,19 @@ theorem isMaximal_of_irreducible [CommRing R] [IsPrincipalIdealRing R] {p : R}
 
 variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
 
-#print PrincipalIdealRing.irreducible_iff_prime /-
+/- warning: principal_ideal_ring.irreducible_iff_prime clashes with gcd_monoid.irreducible_iff_prime -> irreducible_iff_prime
+Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.irreducible_iff_prime irreducible_iff_primeₓ'. -/
+#print irreducible_iff_prime /-
 theorem irreducible_iff_prime {p : R} : Irreducible p ↔ Prime p :=
   ⟨fun hp => (Ideal.span_singleton_prime hp.NeZero).1 <| (isMaximal_of_irreducible hp).IsPrime,
     Prime.irreducible⟩
-#align principal_ideal_ring.irreducible_iff_prime PrincipalIdealRing.irreducible_iff_prime
+#align principal_ideal_ring.irreducible_iff_prime irreducible_iff_prime
 -/
 
-#print PrincipalIdealRing.associates_irreducible_iff_prime /-
+#print associates_irreducible_iff_prime /-
 theorem associates_irreducible_iff_prime : ∀ {p : Associates R}, Irreducible p ↔ Prime p :=
   Associates.irreducible_iff_prime_iff.1 fun _ => irreducible_iff_prime
-#align principal_ideal_ring.associates_irreducible_iff_prime PrincipalIdealRing.associates_irreducible_iff_prime
+#align principal_ideal_ring.associates_irreducible_iff_prime associates_irreducible_iff_prime
 -/
 
 section
@@ -365,7 +367,7 @@ theorem ringHom_mem_submonoid_of_factors_subset_of_units_subset {R S : Type _} [
 /-- A principal ideal domain has unique factorization -/
 instance (priority := 100) to_uniqueFactorizationMonoid : UniqueFactorizationMonoid R :=
   { (IsNoetherianRing.wfDvdMonoid : WfDvdMonoid R) with
-    irreducible_iff_prime := fun _ => PrincipalIdealRing.irreducible_iff_prime }
+    irreducible_iff_prime := fun _ => irreducible_iff_prime }
 #align principal_ideal_ring.to_unique_factorization_monoid PrincipalIdealRing.to_uniqueFactorizationMonoid
 -/
 
@@ -498,7 +500,7 @@ theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
 #print isCoprime_of_prime_dvd /-
 theorem isCoprime_of_prime_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z : R, Prime z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
-  isCoprime_of_irreducible_dvd nonzero fun z zi => H z <| GCDMonoid.prime_of_irreducible zi
+  isCoprime_of_irreducible_dvd nonzero fun z zi => H z <| Irreducible.prime zi
 #align is_coprime_of_prime_dvd isCoprime_of_prime_dvd
 -/
 

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

@@ -50,7 +50,7 @@ section
 variable [Ring R] [AddCommGroup M] [Module R M]
 
 #print Submodule.IsPrincipal /-
-/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:400:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
 /-- An `R`-submodule of `M` is principal if it is generated by one element. -/
 @[mk_iff]
 class Submodule.IsPrincipal (S : Submodule R M) : Prop where

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

@@ -50,7 +50,7 @@ section
 variable [Ring R] [AddCommGroup M] [Module R M]
 
 #print Submodule.IsPrincipal /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
 /-- An `R`-submodule of `M` is principal if it is generated by one element. -/
 @[mk_iff]
 class Submodule.IsPrincipal (S : Submodule R M) : Prop where

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

@@ -475,7 +475,7 @@ theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
 -- this should be proved for UFDs surely?
 theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y :=
   by
-  refine' or_iff_not_imp_left.2 fun h' => _
+  refine' Classical.or_iff_not_imp_left.2 fun h' => _
   apply isCoprime_of_dvd
   · rintro ⟨rfl, rfl⟩; simpa using h
   · rintro z nu nz ⟨w, rfl⟩ dy

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

@@ -3,8 +3,8 @@ Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Morenikeji Neri
 -/
-import Mathbin.Algebra.EuclideanDomain.Instances
-import Mathbin.RingTheory.UniqueFactorizationDomain
+import Algebra.EuclideanDomain.Instances
+import RingTheory.UniqueFactorizationDomain
 
 #align_import ring_theory.principal_ideal_domain from "leanprover-community/mathlib"@"c085f3044fe585c575e322bfab45b3633c48d820"
 
@@ -50,7 +50,7 @@ section
 variable [Ring R] [AddCommGroup M] [Module R M]
 
 #print Submodule.IsPrincipal /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
 /-- An `R`-submodule of `M` is principal if it is generated by one element. -/
 @[mk_iff]
 class Submodule.IsPrincipal (S : Submodule R M) : Prop where

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

@@ -2,15 +2,12 @@
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Morenikeji Neri
-
-! This file was ported from Lean 3 source module ring_theory.principal_ideal_domain
-! leanprover-community/mathlib commit c085f3044fe585c575e322bfab45b3633c48d820
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Algebra.EuclideanDomain.Instances
 import Mathbin.RingTheory.UniqueFactorizationDomain
 
+#align_import ring_theory.principal_ideal_domain from "leanprover-community/mathlib"@"c085f3044fe585c575e322bfab45b3633c48d820"
+
 /-!
 # Principal ideal rings and principal ideal domains
 

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

@@ -53,7 +53,7 @@ section
 variable [Ring R] [AddCommGroup M] [Module R M]
 
 #print Submodule.IsPrincipal /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
 /-- An `R`-submodule of `M` is principal if it is generated by one element. -/
 @[mk_iff]
 class Submodule.IsPrincipal (S : Submodule R M) : Prop where
@@ -61,13 +61,17 @@ class Submodule.IsPrincipal (S : Submodule R M) : Prop where
 #align submodule.is_principal Submodule.IsPrincipal
 -/
 
+#print bot_isPrincipal /-
 instance bot_isPrincipal : (⊥ : Submodule R M).IsPrincipal :=
   ⟨⟨0, by simp⟩⟩
 #align bot_is_principal bot_isPrincipal
+-/
 
+#print top_isPrincipal /-
 instance top_isPrincipal : (⊤ : Submodule R R).IsPrincipal :=
   ⟨⟨1, Ideal.span_singleton_one.symm⟩⟩
 #align top_is_principal top_isPrincipal
+-/
 
 variable (R)
 
@@ -132,9 +136,11 @@ theorem mem_iff_eq_smul_generator (S : Submodule R M) [S.IsPrincipal] {x : M} :
 #align submodule.is_principal.mem_iff_eq_smul_generator Submodule.IsPrincipal.mem_iff_eq_smul_generator
 -/
 
+#print Submodule.IsPrincipal.eq_bot_iff_generator_eq_zero /-
 theorem eq_bot_iff_generator_eq_zero (S : Submodule R M) [S.IsPrincipal] :
     S = ⊥ ↔ generator S = 0 := by rw [← @span_singleton_eq_bot R M, span_singleton_generator]
 #align submodule.is_principal.eq_bot_iff_generator_eq_zero Submodule.IsPrincipal.eq_bot_iff_generator_eq_zero
+-/
 
 end Ring
 
@@ -148,25 +154,31 @@ theorem mem_iff_generator_dvd (S : Ideal R) [S.IsPrincipal] {x : R} : x ∈ S 
 #align submodule.is_principal.mem_iff_generator_dvd Submodule.IsPrincipal.mem_iff_generator_dvd
 -/
 
+#print Submodule.IsPrincipal.prime_generator_of_isPrime /-
 theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_prime : S.IsPrime]
     (ne_bot : S ≠ ⊥) : Prime (generator S) :=
   ⟨fun h => ne_bot ((eq_bot_iff_generator_eq_zero S).2 h), fun h =>
     is_prime.ne_top (S.eq_top_of_isUnit_mem (generator_mem S) h), fun _ _ => by
     simpa only [← mem_iff_generator_dvd S] using is_prime.2⟩
 #align submodule.is_principal.prime_generator_of_is_prime Submodule.IsPrincipal.prime_generator_of_isPrime
+-/
 
+#print Submodule.IsPrincipal.generator_map_dvd_of_mem /-
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
     (hx : x ∈ N) : generator (N.map ϕ) ∣ ϕ x := by rw [← mem_iff_generator_dvd, Submodule.mem_map];
   exact ⟨x, hx, rfl⟩
 #align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_mem
+-/
 
+#print Submodule.IsPrincipal.generator_submoduleImage_dvd_of_mem /-
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
     [(ϕ.submoduleImage N).IsPrincipal] {x : M} (hx : x ∈ N) :
     generator (ϕ.submoduleImage N) ∣ ϕ ⟨x, hNO hx⟩ := by
   rw [← mem_iff_generator_dvd, LinearMap.mem_submoduleImage_of_le hNO]; exact ⟨x, hx, rfl⟩
 #align submodule.is_principal.generator_submodule_image_dvd_of_mem Submodule.IsPrincipal.generator_submoduleImage_dvd_of_mem
+-/
 
 end CommRing
 
@@ -176,6 +188,7 @@ namespace IsPrime
 
 open Submodule.IsPrincipal Ideal
 
+#print IsPrime.to_maximal_ideal /-
 -- TODO -- for a non-ID one could perhaps prove that if p < q are prime then q maximal;
 -- 0 isn't prime in a non-ID PIR but the Krull dimension is still <= 1.
 -- The below result follows from this, but we could also use the below result to
@@ -194,6 +207,7 @@ theorem to_maximal_ideal [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] {S :
       rw [← mul_one (generator S), hy, mul_left_comm, mul_right_inj' this] at hz 
       exact hz.symm ▸ T.mul_mem_right _ (generator_mem T)⟩
 #align is_prime.to_maximal_ideal IsPrime.to_maximal_ideal
+-/
 
 end IsPrime
 
@@ -274,6 +288,7 @@ instance (priority := 100) isNoetherianRing [Ring R] [IsPrincipalIdealRing R] :
 #align principal_ideal_ring.is_noetherian_ring PrincipalIdealRing.isNoetherianRing
 -/
 
+#print PrincipalIdealRing.isMaximal_of_irreducible /-
 theorem isMaximal_of_irreducible [CommRing R] [IsPrincipalIdealRing R] {p : R}
     (hp : Irreducible p) : Ideal.IsMaximal (span R ({p} : Set R)) :=
   ⟨⟨mt Ideal.span_singleton_eq_top.1 hp.1, fun I hI =>
@@ -284,17 +299,22 @@ theorem isMaximal_of_irreducible [CommRing R] [IsPrincipalIdealRing R] {p : R}
       refine' (of_irreducible_mul hp).resolve_right (mt (fun hb => _) (not_le_of_lt hI))
       erw [Ideal.span_singleton_le_span_singleton, IsUnit.mul_right_dvd hb]⟩⟩
 #align principal_ideal_ring.is_maximal_of_irreducible PrincipalIdealRing.isMaximal_of_irreducible
+-/
 
 variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
 
+#print PrincipalIdealRing.irreducible_iff_prime /-
 theorem irreducible_iff_prime {p : R} : Irreducible p ↔ Prime p :=
   ⟨fun hp => (Ideal.span_singleton_prime hp.NeZero).1 <| (isMaximal_of_irreducible hp).IsPrime,
     Prime.irreducible⟩
 #align principal_ideal_ring.irreducible_iff_prime PrincipalIdealRing.irreducible_iff_prime
+-/
 
+#print PrincipalIdealRing.associates_irreducible_iff_prime /-
 theorem associates_irreducible_iff_prime : ∀ {p : Associates R}, Irreducible p ↔ Prime p :=
   Associates.irreducible_iff_prime_iff.1 fun _ => irreducible_iff_prime
 #align principal_ideal_ring.associates_irreducible_iff_prime PrincipalIdealRing.associates_irreducible_iff_prime
+-/
 
 section
 
@@ -307,18 +327,23 @@ noncomputable def factors (a : R) : Multiset R :=
 #align principal_ideal_ring.factors PrincipalIdealRing.factors
 -/
 
+#print PrincipalIdealRing.factors_spec /-
 theorem factors_spec (a : R) (h : a ≠ 0) :
     (∀ b ∈ factors a, Irreducible b) ∧ Associated (factors a).Prod a :=
   by
   unfold factors; rw [dif_neg h]
   exact Classical.choose_spec (WfDvdMonoid.exists_factors a h)
 #align principal_ideal_ring.factors_spec PrincipalIdealRing.factors_spec
+-/
 
+#print PrincipalIdealRing.ne_zero_of_mem_factors /-
 theorem ne_zero_of_mem_factors {R : Type v} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
     {a b : R} (ha : a ≠ 0) (hb : b ∈ factors a) : b ≠ 0 :=
   Irreducible.ne_zero ((factors_spec a ha).1 b hb)
 #align principal_ideal_ring.ne_zero_of_mem_factors PrincipalIdealRing.ne_zero_of_mem_factors
+-/
 
+#print PrincipalIdealRing.mem_submonoid_of_factors_subset_of_units_subset /-
 theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R} (ha : a ≠ 0)
     (hfac : ∀ b ∈ factors a, b ∈ s) (hunit : ∀ c : Rˣ, (c : R) ∈ s) : a ∈ s :=
   by
@@ -326,7 +351,9 @@ theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R
   rw [← hc]
   exact mul_mem (multiset_prod_mem _ hfac) (hunit _)
 #align principal_ideal_ring.mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.mem_submonoid_of_factors_subset_of_units_subset
+-/
 
+#print PrincipalIdealRing.ringHom_mem_submonoid_of_factors_subset_of_units_subset /-
 /-- If a `ring_hom` maps all units and all factors of an element `a` into a submonoid `s`, then it
 also maps `a` into that submonoid. -/
 theorem ringHom_mem_submonoid_of_factors_subset_of_units_subset {R S : Type _} [CommRing R]
@@ -334,6 +361,7 @@ theorem ringHom_mem_submonoid_of_factors_subset_of_units_subset {R S : Type _} [
     (ha : a ≠ 0) (h : ∀ b ∈ factors a, f b ∈ s) (hf : ∀ c : Rˣ, f c ∈ s) : f a ∈ s :=
   mem_submonoid_of_factors_subset_of_units_subset (s.comap f.toMonoidHom) ha h hf
 #align principal_ideal_ring.ring_hom_mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.ringHom_mem_submonoid_of_factors_subset_of_units_subset
+-/
 
 #print PrincipalIdealRing.to_uniqueFactorizationMonoid /-
 -- see Note [lower instance priority]
@@ -356,25 +384,31 @@ variable {S N : Type _} [Ring R] [AddCommGroup M] [AddCommGroup N] [Ring S]
 
 variable [Module R M] [Module R N]
 
+#print Submodule.IsPrincipal.of_comap /-
 theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjective f)
     (S : Submodule R N) [hI : IsPrincipal (S.comap f)] : IsPrincipal S :=
   ⟨⟨f (IsPrincipal.generator (S.comap f)), by
       rw [← Set.image_singleton, ← Submodule.map_span, is_principal.span_singleton_generator,
         Submodule.map_comap_eq_of_surjective hf]⟩⟩
 #align submodule.is_principal.of_comap Submodule.IsPrincipal.of_comap
+-/
 
+#print Ideal.IsPrincipal.of_comap /-
 theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f) (I : Ideal S)
     [hI : IsPrincipal (I.comap f)] : IsPrincipal I :=
   ⟨⟨f (IsPrincipal.generator (I.comap f)), by
       rw [Ideal.submodule_span_eq, ← Set.image_singleton, ← Ideal.map_span,
         Ideal.span_singleton_generator, Ideal.map_comap_of_surjective f hf]⟩⟩
 #align ideal.is_principal.of_comap Ideal.IsPrincipal.of_comap
+-/
 
+#print IsPrincipalIdealRing.of_surjective /-
 /-- The surjective image of a principal ideal ring is again a principal ideal ring. -/
 theorem IsPrincipalIdealRing.of_surjective [IsPrincipalIdealRing R] (f : R →+* S)
     (hf : Function.Surjective f) : IsPrincipalIdealRing S :=
   ⟨fun I => Ideal.IsPrincipal.of_comap f hf I⟩
 #align is_principal_ideal_ring.of_surjective IsPrincipalIdealRing.of_surjective
+-/
 
 end Surjective
 
@@ -408,20 +442,27 @@ theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
 #align span_gcd span_gcd
 -/
 
+#print gcd_dvd_iff_exists /-
 theorem gcd_dvd_iff_exists (a b : R) {z} : gcd a b ∣ z ↔ ∃ x y, z = a * x + b * y := by
   simp_rw [mul_comm a, mul_comm b, @eq_comm _ z, ← Ideal.mem_span_pair, ← span_gcd,
     Ideal.mem_span_singleton]
 #align gcd_dvd_iff_exists gcd_dvd_iff_exists
+-/
 
+#print exists_gcd_eq_mul_add_mul /-
 /-- **Bézout's lemma** -/
 theorem exists_gcd_eq_mul_add_mul (a b : R) : ∃ x y, gcd a b = a * x + b * y := by
   rw [← gcd_dvd_iff_exists]
 #align exists_gcd_eq_mul_add_mul exists_gcd_eq_mul_add_mul
+-/
 
+#print gcd_isUnit_iff /-
 theorem gcd_isUnit_iff (x y : R) : IsUnit (gcd x y) ↔ IsCoprime x y := by
   rw [IsCoprime, ← Ideal.mem_span_pair, ← span_gcd, ← span_singleton_eq_top, eq_top_iff_one]
 #align gcd_is_unit_iff gcd_isUnit_iff
+-/
 
+#print isCoprime_of_dvd /-
 -- this should be proved for UFDs surely?
 theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z ∈ nonunits R, z ≠ 0 → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
@@ -431,7 +472,9 @@ theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
   refine' H _ h _ (gcd_dvd_left _ _) (gcd_dvd_right _ _)
   rwa [Ne, gcd_eq_zero_iff]
 #align is_coprime_of_dvd isCoprime_of_dvd
+-/
 
+#print dvd_or_coprime /-
 -- this should be proved for UFDs surely?
 theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y :=
   by
@@ -442,7 +485,9 @@ theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y
     refine' h' (dvd_trans _ dy)
     simpa using mul_dvd_mul_left z (isUnit_iff_dvd_one.1 <| (of_irreducible_mul h).resolve_left nu)
 #align dvd_or_coprime dvd_or_coprime
+-/
 
+#print isCoprime_of_irreducible_dvd /-
 theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z : R, Irreducible z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
   by
@@ -451,12 +496,16 @@ theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
   obtain ⟨i, h1, h2⟩ := WfDvdMonoid.exists_irreducible_factor znu znz
   apply H i h1 <;> · apply dvd_trans h2; assumption
 #align is_coprime_of_irreducible_dvd isCoprime_of_irreducible_dvd
+-/
 
+#print isCoprime_of_prime_dvd /-
 theorem isCoprime_of_prime_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z : R, Prime z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
   isCoprime_of_irreducible_dvd nonzero fun z zi => H z <| GCDMonoid.prime_of_irreducible zi
 #align is_coprime_of_prime_dvd isCoprime_of_prime_dvd
+-/
 
+#print Irreducible.coprime_iff_not_dvd /-
 theorem Irreducible.coprime_iff_not_dvd {p n : R} (pp : Irreducible p) : IsCoprime p n ↔ ¬p ∣ n :=
   by
   constructor
@@ -473,6 +522,7 @@ theorem Irreducible.coprime_iff_not_dvd {p n : R} (pp : Irreducible p) : IsCopri
     rintro z zi zp zn
     exact nd ((zi.associated_of_dvd pp zp).symm.Dvd.trans zn)
 #align irreducible.coprime_iff_not_dvd Irreducible.coprime_iff_not_dvd
+-/
 
 #print Prime.coprime_iff_not_dvd /-
 theorem Prime.coprime_iff_not_dvd {p n : R} (pp : Prime p) : IsCoprime p n ↔ ¬p ∣ n :=
@@ -480,23 +530,31 @@ theorem Prime.coprime_iff_not_dvd {p n : R} (pp : Prime p) : IsCoprime p n ↔ 
 #align prime.coprime_iff_not_dvd Prime.coprime_iff_not_dvd
 -/
 
+#print Irreducible.dvd_iff_not_coprime /-
 theorem Irreducible.dvd_iff_not_coprime {p n : R} (hp : Irreducible p) : p ∣ n ↔ ¬IsCoprime p n :=
   iff_not_comm.2 hp.coprime_iff_not_dvd
 #align irreducible.dvd_iff_not_coprime Irreducible.dvd_iff_not_coprime
+-/
 
+#print Irreducible.coprime_pow_of_not_dvd /-
 theorem Irreducible.coprime_pow_of_not_dvd {p a : R} (m : ℕ) (hp : Irreducible p) (h : ¬p ∣ a) :
     IsCoprime a (p ^ m) :=
   (hp.coprime_iff_not_dvd.2 h).symm.pow_right
 #align irreducible.coprime_pow_of_not_dvd Irreducible.coprime_pow_of_not_dvd
+-/
 
+#print Irreducible.coprime_or_dvd /-
 theorem Irreducible.coprime_or_dvd {p : R} (hp : Irreducible p) (i : R) : IsCoprime p i ∨ p ∣ i :=
   (em _).imp_right hp.dvd_iff_not_coprime.2
 #align irreducible.coprime_or_dvd Irreducible.coprime_or_dvd
+-/
 
+#print exists_associated_pow_of_mul_eq_pow' /-
 theorem exists_associated_pow_of_mul_eq_pow' {a b c : R} (hab : IsCoprime a b) {k : ℕ}
     (h : a * b = c ^ k) : ∃ d, Associated (d ^ k) a :=
   exists_associated_pow_of_mul_eq_pow ((gcd_isUnit_iff _ _).mpr hab) h
 #align exists_associated_pow_of_mul_eq_pow' exists_associated_pow_of_mul_eq_pow'
+-/
 
 end
 
@@ -527,6 +585,7 @@ theorem nonPrincipals_eq_empty_iff : nonPrincipals R = ∅ ↔ IsPrincipalIdealR
 #align non_principals_eq_empty_iff nonPrincipals_eq_empty_iff
 -/
 
+#print nonPrincipals_zorn /-
 /-- Any chain in the set of non-principal ideals has an upper bound which is non-principal.
 (Namely, the union of the chain is such an upper bound.)
 -/
@@ -543,6 +602,7 @@ theorem nonPrincipals_zorn (c : Set (Ideal R)) (hs : c ⊆ nonPrincipals R)
   rw [← hSupJ, hx, nonPrincipals_def] at hs 
   exact hs ⟨⟨x, rfl⟩⟩
 #align non_principals_zorn nonPrincipals_zorn
+-/
 
 #print IsPrincipalIdealRing.of_prime /-
 /-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/

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

@@ -53,7 +53,7 @@ section
 variable [Ring R] [AddCommGroup M] [Module R M]
 
 #print Submodule.IsPrincipal /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
 /-- An `R`-submodule of `M` is principal if it is generated by one element. -/
 @[mk_iff]
 class Submodule.IsPrincipal (S : Submodule R M) : Prop where
@@ -214,24 +214,23 @@ theorem mod_mem_iff {S : Ideal R} {x y : R} (hy : y ∈ S) : x % y ∈ S ↔ x 
 -- see Note [lower instance priority]
 instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincipalIdealRing R
     where principal S :=
-    ⟨if h : { x : R | x ∈ S ∧ x ≠ 0 }.Nonempty then
+    ⟨if h : {x : R | x ∈ S ∧ x ≠ 0}.Nonempty then
         have wf : WellFounded (EuclideanDomain.r : R → R → Prop) := EuclideanDomain.r_wellFounded
         have hmin :
-          WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h ∈ S ∧
-            WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h ≠ 0 :=
-          WellFounded.min_mem wf { x : R | x ∈ S ∧ x ≠ 0 } h
-        ⟨WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h,
+          WellFounded.min wf {x : R | x ∈ S ∧ x ≠ 0} h ∈ S ∧
+            WellFounded.min wf {x : R | x ∈ S ∧ x ≠ 0} h ≠ 0 :=
+          WellFounded.min_mem wf {x : R | x ∈ S ∧ x ≠ 0} h
+        ⟨WellFounded.min wf {x : R | x ∈ S ∧ x ≠ 0} h,
           Submodule.ext fun x =>
             ⟨fun hx =>
-              div_add_mod x (WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h) ▸
+              div_add_mod x (WellFounded.min wf {x : R | x ∈ S ∧ x ≠ 0} h) ▸
                 (Ideal.mem_span_singleton.2 <|
                   dvd_add (dvd_mul_right _ _) <|
                     by
                     have :
-                      x % WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h ∉
-                        { x : R | x ∈ S ∧ x ≠ 0 } :=
+                      x % WellFounded.min wf {x : R | x ∈ S ∧ x ≠ 0} h ∉ {x : R | x ∈ S ∧ x ≠ 0} :=
                       fun h₁ => WellFounded.not_lt_min wf _ h h₁ (mod_lt x hmin.2)
-                    have : x % WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h = 0 :=
+                    have : x % WellFounded.min wf {x : R | x ∈ S ∧ x ≠ 0} h = 0 :=
                       by
                       simp only [not_and_or, Set.mem_setOf_eq, not_ne_iff] at this 
                       cases this; cases this ((mod_mem_iff hmin.1).2 hx); exact this
@@ -510,7 +509,7 @@ variable (R) [CommRing R]
 #print nonPrincipals /-
 /-- `non_principals R` is the set of all ideals of `R` that are not principal ideals. -/
 def nonPrincipals :=
-  { I : Ideal R | ¬I.IsPrincipal }
+  {I : Ideal R | ¬I.IsPrincipal}
 #align non_principals nonPrincipals
 -/
 

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

@@ -191,7 +191,7 @@ theorem to_maximal_ideal [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] {S :
         exact (hxS <| hTS hxT).elim
       cases' (mem_iff_generator_dvd _).1 h with y hy
       have : generator S ≠ 0 := mt (eq_bot_iff_generator_eq_zero _).2 hS
-      rw [← mul_one (generator S), hy, mul_left_comm, mul_right_inj' this] at hz
+      rw [← mul_one (generator S), hy, mul_left_comm, mul_right_inj' this] at hz 
       exact hz.symm ▸ T.mul_mem_right _ (generator_mem T)⟩
 #align is_prime.to_maximal_ideal IsPrime.to_maximal_ideal
 
@@ -233,7 +233,7 @@ instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincip
                       fun h₁ => WellFounded.not_lt_min wf _ h h₁ (mod_lt x hmin.2)
                     have : x % WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h = 0 :=
                       by
-                      simp only [not_and_or, Set.mem_setOf_eq, not_ne_iff] at this
+                      simp only [not_and_or, Set.mem_setOf_eq, not_ne_iff] at this 
                       cases this; cases this ((mod_mem_iff hmin.1).2 hx); exact this
                     simp [*]),
               fun hx =>
@@ -389,7 +389,7 @@ variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [GCDMonoid R]
 theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
   by
   obtain ⟨d, hd⟩ := IsPrincipalIdealRing.principal (span ({x, y} : Set R))
-  rw [submodule_span_eq] at hd
+  rw [submodule_span_eq] at hd 
   rw [hd]
   suffices Associated d (gcd x y) by
     obtain ⟨D, HD⟩ := this
@@ -405,7 +405,7 @@ theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
   · obtain ⟨r, s, rfl⟩ : ∃ r s, r * x + s * y = d := by
       rw [← Ideal.mem_span_pair, hd, Ideal.mem_span_singleton]
     apply dvd_add <;> apply dvd_mul_of_dvd_right
-    exacts[gcd_dvd_left x y, gcd_dvd_right x y]
+    exacts [gcd_dvd_left x y, gcd_dvd_right x y]
 #align span_gcd span_gcd
 -/
 
@@ -541,7 +541,7 @@ theorem nonPrincipals_zorn (c : Set (Ideal R)) (hs : c ⊆ nonPrincipals R)
   obtain ⟨J, hJc, hxJ⟩ := (Submodule.mem_sSup_of_directed ⟨K, hKmem⟩ hchain.directed_on).1 hxmem
   have hSupJ : Sup c = J := le_antisymm (by simp [hx, Ideal.span_le, hxJ]) (le_sSup hJc)
   specialize hs hJc
-  rw [← hSupJ, hx, nonPrincipals_def] at hs
+  rw [← hSupJ, hx, nonPrincipals_def] at hs 
   exact hs ⟨⟨x, rfl⟩⟩
 #align non_principals_zorn nonPrincipals_zorn
 
@@ -582,14 +582,14 @@ theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPr
       ⟨⟨a * b, (le_antisymm fun i hi => _) <| (span_singleton_mul_span_singleton a b).ge.trans _⟩⟩
   · have hisup : i ∈ I ⊔ span {y} := Ideal.mem_sup_left hi
     have : y ∈ I ⊔ span {y} := Ideal.mem_sup_right (Ideal.mem_span_singleton_self y)
-    erw [ha, mem_span_singleton'] at hisup this
+    erw [ha, mem_span_singleton'] at hisup this 
     obtain ⟨v, rfl⟩ := this
     obtain ⟨u, rfl⟩ := hisup
     have hucolon : u ∈ I.colon (span {v * a}) :=
       by
       rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
       exact mul_mem_right _ _ hi
-    erw [hb, mem_span_singleton'] at hucolon
+    erw [hb, mem_span_singleton'] at hucolon 
     obtain ⟨z, rfl⟩ := hucolon
     exact mem_span_singleton'.2 ⟨z, by ring⟩
   · rw [← Ideal.submodule_span_eq, ← ha, Ideal.sup_mul, sup_le_iff,

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

@@ -46,7 +46,7 @@ open Set Function
 
 open Submodule
 
-open Classical
+open scoped Classical
 
 section
 
@@ -299,7 +299,7 @@ theorem associates_irreducible_iff_prime : ∀ {p : Associates R}, Irreducible p
 
 section
 
-open Classical
+open scoped Classical
 
 #print PrincipalIdealRing.factors /-
 /-- `factors a` is a multiset of irreducible elements whose product is `a`, up to units -/

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

@@ -61,22 +61,10 @@ class Submodule.IsPrincipal (S : Submodule R M) : Prop where
 #align submodule.is_principal Submodule.IsPrincipal
 -/
 
-/- warning: bot_is_principal -> bot_isPrincipal is a dubious translation:
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-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align bot_is_principal bot_isPrincipalₓ'. -/
 instance bot_isPrincipal : (⊥ : Submodule R M).IsPrincipal :=
   ⟨⟨0, by simp⟩⟩
 #align bot_is_principal bot_isPrincipal
 
-/- warning: top_is_principal -> top_isPrincipal is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R], Submodule.IsPrincipal.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Top.top.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Submodule.instTopSubmodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))))
-Case conversion may be inaccurate. Consider using '#align top_is_principal top_isPrincipalₓ'. -/
 instance top_isPrincipal : (⊤ : Submodule R R).IsPrincipal :=
   ⟨⟨1, Ideal.span_singleton_one.symm⟩⟩
 #align top_is_principal top_isPrincipal
@@ -144,12 +132,6 @@ theorem mem_iff_eq_smul_generator (S : Submodule R M) [S.IsPrincipal] {x : M} :
 #align submodule.is_principal.mem_iff_eq_smul_generator Submodule.IsPrincipal.mem_iff_eq_smul_generator
 -/
 
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-but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : Ring.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) [_inst_4 : Submodule.IsPrincipal.{u1, u2} R M _inst_2 _inst_1 _inst_3 S], Iff (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) S (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.instBotSubmodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))) (Eq.{succ u2} M (Submodule.IsPrincipal.generator.{u1, u2} R M _inst_1 _inst_2 _inst_3 S _inst_4) (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_1))))))))
-Case conversion may be inaccurate. Consider using '#align submodule.is_principal.eq_bot_iff_generator_eq_zero Submodule.IsPrincipal.eq_bot_iff_generator_eq_zeroₓ'. -/
 theorem eq_bot_iff_generator_eq_zero (S : Submodule R M) [S.IsPrincipal] :
     S = ⊥ ↔ generator S = 0 := by rw [← @span_singleton_eq_bot R M, span_singleton_generator]
 #align submodule.is_principal.eq_bot_iff_generator_eq_zero Submodule.IsPrincipal.eq_bot_iff_generator_eq_zero
@@ -166,12 +148,6 @@ theorem mem_iff_generator_dvd (S : Ideal R) [S.IsPrincipal] {x : R} : x ∈ S 
 #align submodule.is_principal.mem_iff_generator_dvd Submodule.IsPrincipal.mem_iff_generator_dvd
 -/
 
-/- warning: submodule.is_principal.prime_generator_of_is_prime -> Submodule.IsPrincipal.prime_generator_of_isPrime is a dubious translation:
-lean 3 declaration is
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-  forall {R : Type.{u1}} [_inst_2 : CommRing.{u1} R] (S : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) S] [is_prime : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) S], (Ne.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) S (Bot.bot.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.instBotSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))))) -> (Prime.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) S _inst_4))
-Case conversion may be inaccurate. Consider using '#align submodule.is_principal.prime_generator_of_is_prime Submodule.IsPrincipal.prime_generator_of_isPrimeₓ'. -/
 theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_prime : S.IsPrime]
     (ne_bot : S ≠ ⊥) : Prime (generator S) :=
   ⟨fun h => ne_bot ((eq_bot_iff_generator_eq_zero S).2 h), fun h =>
@@ -179,18 +155,12 @@ theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_p
     simpa only [← mem_iff_generator_dvd S] using is_prime.2⟩
 #align submodule.is_principal.prime_generator_of_is_prime Submodule.IsPrincipal.prime_generator_of_isPrime
 
-/- warning: submodule.is_principal.generator_map_dvd_of_mem -> Submodule.IsPrincipal.generator_map_dvd_of_mem is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
     (hx : x ∈ N) : generator (N.map ϕ) ∣ ϕ x := by rw [← mem_iff_generator_dvd, Submodule.mem_map];
   exact ⟨x, hx, rfl⟩
 #align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_mem
 
-/- warning: submodule.is_principal.generator_submodule_image_dvd_of_mem -> Submodule.IsPrincipal.generator_submoduleImage_dvd_of_mem is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_submodule_image_dvd_of_mem Submodule.IsPrincipal.generator_submoduleImage_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
     [(ϕ.submoduleImage N).IsPrincipal] {x : M} (hx : x ∈ N) :
@@ -206,12 +176,6 @@ namespace IsPrime
 
 open Submodule.IsPrincipal Ideal
 
-/- warning: is_prime.to_maximal_ideal -> IsPrime.to_maximal_ideal is a dubious translation:
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-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {S : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} [hpi : Ideal.IsPrime.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) S], (Ne.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) S (Bot.bot.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasBot.{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.IsMaximal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) S)
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-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {S : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} [hpi : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) S], (Ne.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) S (Bot.bot.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.instBotSubmodule.{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)))))) -> (Ideal.IsMaximal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) S)
-Case conversion may be inaccurate. Consider using '#align is_prime.to_maximal_ideal IsPrime.to_maximal_idealₓ'. -/
 -- TODO -- for a non-ID one could perhaps prove that if p < q are prime then q maximal;
 -- 0 isn't prime in a non-ID PIR but the Krull dimension is still <= 1.
 -- The below result follows from this, but we could also use the below result to
@@ -311,12 +275,6 @@ instance (priority := 100) isNoetherianRing [Ring R] [IsPrincipalIdealRing R] :
 #align principal_ideal_ring.is_noetherian_ring PrincipalIdealRing.isNoetherianRing
 -/
 
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-Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.is_maximal_of_irreducible PrincipalIdealRing.isMaximal_of_irreducibleₓ'. -/
 theorem isMaximal_of_irreducible [CommRing R] [IsPrincipalIdealRing R] {p : R}
     (hp : Irreducible p) : Ideal.IsMaximal (span R ({p} : Set R)) :=
   ⟨⟨mt Ideal.span_singleton_eq_top.1 hp.1, fun I hI =>
@@ -330,23 +288,11 @@ theorem isMaximal_of_irreducible [CommRing R] [IsPrincipalIdealRing R] {p : R}
 
 variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
 
-/- warning: principal_ideal_ring.irreducible_iff_prime -> PrincipalIdealRing.irreducible_iff_prime is a dubious translation:
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-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, Iff (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) (Prime.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) p)
-but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, Iff (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) p) (Prime.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)) p)
-Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.irreducible_iff_prime PrincipalIdealRing.irreducible_iff_primeₓ'. -/
 theorem irreducible_iff_prime {p : R} : Irreducible p ↔ Prime p :=
   ⟨fun hp => (Ideal.span_singleton_prime hp.NeZero).1 <| (isMaximal_of_irreducible hp).IsPrime,
     Prime.irreducible⟩
 #align principal_ideal_ring.irreducible_iff_prime PrincipalIdealRing.irreducible_iff_prime
 
-/- warning: principal_ideal_ring.associates_irreducible_iff_prime -> PrincipalIdealRing.associates_irreducible_iff_prime is a dubious translation:
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-Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.associates_irreducible_iff_prime PrincipalIdealRing.associates_irreducible_iff_primeₓ'. -/
 theorem associates_irreducible_iff_prime : ∀ {p : Associates R}, Irreducible p ↔ Prime p :=
   Associates.irreducible_iff_prime_iff.1 fun _ => irreducible_iff_prime
 #align principal_ideal_ring.associates_irreducible_iff_prime PrincipalIdealRing.associates_irreducible_iff_prime
@@ -362,12 +308,6 @@ noncomputable def factors (a : R) : Multiset R :=
 #align principal_ideal_ring.factors PrincipalIdealRing.factors
 -/
 
-/- warning: principal_ideal_ring.factors_spec -> PrincipalIdealRing.factors_spec is a dubious translation:
-lean 3 declaration is
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-Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.factors_spec PrincipalIdealRing.factors_specₓ'. -/
 theorem factors_spec (a : R) (h : a ≠ 0) :
     (∀ b ∈ factors a, Irreducible b) ∧ Associated (factors a).Prod a :=
   by
@@ -375,23 +315,11 @@ theorem factors_spec (a : R) (h : a ≠ 0) :
   exact Classical.choose_spec (WfDvdMonoid.exists_factors a h)
 #align principal_ideal_ring.factors_spec PrincipalIdealRing.factors_spec
 
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 theorem ne_zero_of_mem_factors {R : Type v} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
     {a b : R} (ha : a ≠ 0) (hb : b ∈ factors a) : b ≠ 0 :=
   Irreducible.ne_zero ((factors_spec a ha).1 b hb)
 #align principal_ideal_ring.ne_zero_of_mem_factors PrincipalIdealRing.ne_zero_of_mem_factors
 
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 theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R} (ha : a ≠ 0)
     (hfac : ∀ b ∈ factors a, b ∈ s) (hunit : ∀ c : Rˣ, (c : R) ∈ s) : a ∈ s :=
   by
@@ -400,9 +328,6 @@ theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R
   exact mul_mem (multiset_prod_mem _ hfac) (hunit _)
 #align principal_ideal_ring.mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.mem_submonoid_of_factors_subset_of_units_subset
 
-/- warning: principal_ideal_ring.ring_hom_mem_submonoid_of_factors_subset_of_units_subset -> PrincipalIdealRing.ringHom_mem_submonoid_of_factors_subset_of_units_subset is a dubious translation:
-<too large>
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 /-- If a `ring_hom` maps all units and all factors of an element `a` into a submonoid `s`, then it
 also maps `a` into that submonoid. -/
 theorem ringHom_mem_submonoid_of_factors_subset_of_units_subset {R S : Type _} [CommRing R]
@@ -432,12 +357,6 @@ variable {S N : Type _} [Ring R] [AddCommGroup M] [AddCommGroup N] [Ring S]
 
 variable [Module R M] [Module R N]
 
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 theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjective f)
     (S : Submodule R N) [hI : IsPrincipal (S.comap f)] : IsPrincipal S :=
   ⟨⟨f (IsPrincipal.generator (S.comap f)), by
@@ -445,12 +364,6 @@ theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjec
         Submodule.map_comap_eq_of_surjective hf]⟩⟩
 #align submodule.is_principal.of_comap Submodule.IsPrincipal.of_comap
 
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 theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f) (I : Ideal S)
     [hI : IsPrincipal (I.comap f)] : IsPrincipal I :=
   ⟨⟨f (IsPrincipal.generator (I.comap f)), by
@@ -458,12 +371,6 @@ theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f)
         Ideal.span_singleton_generator, Ideal.map_comap_of_surjective f hf]⟩⟩
 #align ideal.is_principal.of_comap Ideal.IsPrincipal.of_comap
 
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 /-- The surjective image of a principal ideal ring is again a principal ideal ring. -/
 theorem IsPrincipalIdealRing.of_surjective [IsPrincipalIdealRing R] (f : R →+* S)
     (hf : Function.Surjective f) : IsPrincipalIdealRing S :=
@@ -502,44 +409,20 @@ theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
 #align span_gcd span_gcd
 -/
 
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 theorem gcd_dvd_iff_exists (a b : R) {z} : gcd a b ∣ z ↔ ∃ x y, z = a * x + b * y := by
   simp_rw [mul_comm a, mul_comm b, @eq_comm _ z, ← Ideal.mem_span_pair, ← span_gcd,
     Ideal.mem_span_singleton]
 #align gcd_dvd_iff_exists gcd_dvd_iff_exists
 
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 /-- **Bézout's lemma** -/
 theorem exists_gcd_eq_mul_add_mul (a b : R) : ∃ x y, gcd a b = a * x + b * y := by
   rw [← gcd_dvd_iff_exists]
 #align exists_gcd_eq_mul_add_mul exists_gcd_eq_mul_add_mul
 
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 theorem gcd_isUnit_iff (x y : R) : IsUnit (gcd x y) ↔ IsCoprime x y := by
   rw [IsCoprime, ← Ideal.mem_span_pair, ← span_gcd, ← span_singleton_eq_top, eq_top_iff_one]
 #align gcd_is_unit_iff gcd_isUnit_iff
 
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 -- this should be proved for UFDs surely?
 theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z ∈ nonunits R, z ≠ 0 → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
@@ -550,12 +433,6 @@ theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
   rwa [Ne, gcd_eq_zero_iff]
 #align is_coprime_of_dvd isCoprime_of_dvd
 
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 -- this should be proved for UFDs surely?
 theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y :=
   by
@@ -567,12 +444,6 @@ theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y
     simpa using mul_dvd_mul_left z (isUnit_iff_dvd_one.1 <| (of_irreducible_mul h).resolve_left nu)
 #align dvd_or_coprime dvd_or_coprime
 
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 theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z : R, Irreducible z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
   by
@@ -582,23 +453,11 @@ theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
   apply H i h1 <;> · apply dvd_trans h2; assumption
 #align is_coprime_of_irreducible_dvd isCoprime_of_irreducible_dvd
 
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 theorem isCoprime_of_prime_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z : R, Prime z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
   isCoprime_of_irreducible_dvd nonzero fun z zi => H z <| GCDMonoid.prime_of_irreducible zi
 #align is_coprime_of_prime_dvd isCoprime_of_prime_dvd
 
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 theorem Irreducible.coprime_iff_not_dvd {p n : R} (pp : Irreducible p) : IsCoprime p n ↔ ¬p ∣ n :=
   by
   constructor
@@ -622,43 +481,19 @@ theorem Prime.coprime_iff_not_dvd {p n : R} (pp : Prime p) : IsCoprime p n ↔ 
 #align prime.coprime_iff_not_dvd Prime.coprime_iff_not_dvd
 -/
 
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 theorem Irreducible.dvd_iff_not_coprime {p n : R} (hp : Irreducible p) : p ∣ n ↔ ¬IsCoprime p n :=
   iff_not_comm.2 hp.coprime_iff_not_dvd
 #align irreducible.dvd_iff_not_coprime Irreducible.dvd_iff_not_coprime
 
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 theorem Irreducible.coprime_pow_of_not_dvd {p a : R} (m : ℕ) (hp : Irreducible p) (h : ¬p ∣ a) :
     IsCoprime a (p ^ m) :=
   (hp.coprime_iff_not_dvd.2 h).symm.pow_right
 #align irreducible.coprime_pow_of_not_dvd Irreducible.coprime_pow_of_not_dvd
 
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 theorem Irreducible.coprime_or_dvd {p : R} (hp : Irreducible p) (i : R) : IsCoprime p i ∨ p ∣ i :=
   (em _).imp_right hp.dvd_iff_not_coprime.2
 #align irreducible.coprime_or_dvd Irreducible.coprime_or_dvd
 
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 theorem exists_associated_pow_of_mul_eq_pow' {a b c : R} (hab : IsCoprime a b) {k : ℕ}
     (h : a * b = c ^ k) : ∃ d, Associated (d ^ k) a :=
   exists_associated_pow_of_mul_eq_pow ((gcd_isUnit_iff _ _).mpr hab) h
@@ -693,12 +528,6 @@ theorem nonPrincipals_eq_empty_iff : nonPrincipals R = ∅ ↔ IsPrincipalIdealR
 #align non_principals_eq_empty_iff nonPrincipals_eq_empty_iff
 -/
 
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-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4546 : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4548 : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4546 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4548) c) -> (forall {K : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
-Case conversion may be inaccurate. Consider using '#align non_principals_zorn nonPrincipals_zornₓ'. -/
 /-- Any chain in the set of non-principal ideals has an upper bound which is non-principal.
 (Namely, the union of the chain is such an upper bound.)
 -/

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

@@ -132,10 +132,8 @@ theorem Ideal.span_singleton_generator (I : Ideal R) [I.IsPrincipal] :
 
 #print Submodule.IsPrincipal.generator_mem /-
 @[simp]
-theorem generator_mem (S : Submodule R M) [S.IsPrincipal] : generator S ∈ S :=
-  by
-  conv_rhs => rw [← span_singleton_generator S]
-  exact subset_span (mem_singleton _)
+theorem generator_mem (S : Submodule R M) [S.IsPrincipal] : generator S ∈ S := by
+  conv_rhs => rw [← span_singleton_generator S]; exact subset_span (mem_singleton _)
 #align submodule.is_principal.generator_mem Submodule.IsPrincipal.generator_mem
 -/
 
@@ -186,9 +184,7 @@ theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_p
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
-    (hx : x ∈ N) : generator (N.map ϕ) ∣ ϕ x :=
-  by
-  rw [← mem_iff_generator_dvd, Submodule.mem_map]
+    (hx : x ∈ N) : generator (N.map ϕ) ∣ ϕ x := by rw [← mem_iff_generator_dvd, Submodule.mem_map];
   exact ⟨x, hx, rfl⟩
 #align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_mem
 
@@ -198,10 +194,8 @@ Case conversion may be inaccurate. Consider using '#align submodule.is_principal
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
     [(ϕ.submoduleImage N).IsPrincipal] {x : M} (hx : x ∈ N) :
-    generator (ϕ.submoduleImage N) ∣ ϕ ⟨x, hNO hx⟩ :=
-  by
-  rw [← mem_iff_generator_dvd, LinearMap.mem_submoduleImage_of_le hNO]
-  exact ⟨x, hx, rfl⟩
+    generator (ϕ.submoduleImage N) ∣ ϕ ⟨x, hNO hx⟩ := by
+  rw [← mem_iff_generator_dvd, LinearMap.mem_submoduleImage_of_le hNO]; exact ⟨x, hx, rfl⟩
 #align submodule.is_principal.generator_submodule_image_dvd_of_mem Submodule.IsPrincipal.generator_submoduleImage_dvd_of_mem
 
 end CommRing
@@ -229,8 +223,7 @@ theorem to_maximal_ideal [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] {S :
       intro T x hST hxS hxT
       cases' (mem_iff_generator_dvd _).1 (hST <| generator_mem S) with z hz
       cases hpi.mem_or_mem (show generator T * z ∈ S from hz ▸ generator_mem S)
-      · have hTS : T ≤ S
-        rwa [← T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]
+      · have hTS : T ≤ S; rwa [← T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]
         exact (hxS <| hTS hxT).elim
       cases' (mem_iff_generator_dvd _).1 h with y hy
       have : generator S ≠ 0 := mt (eq_bot_iff_generator_eq_zero _).2 hS
@@ -277,9 +270,7 @@ instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincip
                     have : x % WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h = 0 :=
                       by
                       simp only [not_and_or, Set.mem_setOf_eq, not_ne_iff] at this
-                      cases this
-                      cases this ((mod_mem_iff hmin.1).2 hx)
-                      exact this
+                      cases this; cases this ((mod_mem_iff hmin.1).2 hx); exact this
                     simp [*]),
               fun hx =>
               let ⟨y, hy⟩ := Ideal.mem_span_singleton.1 hx
@@ -570,8 +561,7 @@ theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y
   by
   refine' or_iff_not_imp_left.2 fun h' => _
   apply isCoprime_of_dvd
-  · rintro ⟨rfl, rfl⟩
-    simpa using h
+  · rintro ⟨rfl, rfl⟩; simpa using h
   · rintro z nu nz ⟨w, rfl⟩ dy
     refine' h' (dvd_trans _ dy)
     simpa using mul_dvd_mul_left z (isUnit_iff_dvd_one.1 <| (of_irreducible_mul h).resolve_left nu)
@@ -589,9 +579,7 @@ theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
   apply isCoprime_of_dvd x y nonzero
   intro z znu znz zx zy
   obtain ⟨i, h1, h2⟩ := WfDvdMonoid.exists_irreducible_factor znu znz
-  apply H i h1 <;>
-    · apply dvd_trans h2
-      assumption
+  apply H i h1 <;> · apply dvd_trans h2; assumption
 #align is_coprime_of_irreducible_dvd isCoprime_of_irreducible_dvd
 
 /- warning: is_coprime_of_prime_dvd -> isCoprime_of_prime_dvd is a dubious translation:

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

@@ -182,10 +182,7 @@ theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_p
 #align submodule.is_principal.prime_generator_of_is_prime Submodule.IsPrincipal.prime_generator_of_isPrime
 
 /- warning: submodule.is_principal.generator_map_dvd_of_mem -> Submodule.IsPrincipal.generator_map_dvd_of_mem is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R 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+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
@@ -196,10 +193,7 @@ theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.ma
 #align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_mem
 
 /- warning: submodule.is_principal.generator_submodule_image_dvd_of_mem -> Submodule.IsPrincipal.generator_submoduleImage_dvd_of_mem is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_submodule_image_dvd_of_mem Submodule.IsPrincipal.generator_submoduleImage_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
@@ -416,10 +410,7 @@ theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R
 #align principal_ideal_ring.mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.mem_submonoid_of_factors_subset_of_units_subset
 
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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) c)) s) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) a) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f a) s)
+<too large>
 Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.ring_hom_mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.ringHom_mem_submonoid_of_factors_subset_of_units_subsetₓ'. -/
 /-- If a `ring_hom` maps all units and all factors of an element `a` into a submonoid `s`, then it
 also maps `a` into that submonoid. -/

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

@@ -185,7 +185,7 @@ theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_p
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N) _inst_4) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) => M -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ x))
 but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : M) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ x))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : M) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ x))
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
@@ -199,7 +199,7 @@ theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.ma
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N) _inst_4) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) => (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
 but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R 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(LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R 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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ (Subtype.mk.{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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R 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(SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : 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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => R) _x) 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R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_submodule_image_dvd_of_mem Submodule.IsPrincipal.generator_submoduleImage_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
@@ -454,7 +454,7 @@ variable [Module R M] [Module R N]
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} N] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_6 : Module.{u1, u3} R N (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3)] (f : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u2, succ u3} M N (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) (fun (_x : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) => M -> N) (LinearMap.hasCoeToFun.{u1, u1, u2, u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) -> (forall (S : Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u1, u2} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u1, u1, u2, u3, max u2 u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f S)], Submodule.IsPrincipal.{u1, u3} R N _inst_1 _inst_3 _inst_6 S)
 but is expected to have type
-  forall {R : Type.{u2}} {M : Type.{u3}} {N : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u1} N] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_6 : Module.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3)] (f : LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u3, succ u1} M N (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : M) => N) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) -> (forall (S : Submodule.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u2, u3} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u2, u2, u3, u1, max u3 u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f S)], Submodule.IsPrincipal.{u2, u1} R N _inst_1 _inst_3 _inst_6 S)
+  forall {R : Type.{u2}} {M : Type.{u3}} {N : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u1} N] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_6 : Module.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3)] (f : LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u3, succ u1} M N (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6193 : M) => N) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) -> (forall (S : Submodule.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u2, u3} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u2, u2, u3, u1, max u3 u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f S)], Submodule.IsPrincipal.{u2, u1} R N _inst_1 _inst_3 _inst_6 S)
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.of_comap Submodule.IsPrincipal.of_comapₓ'. -/
 theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjective f)
     (S : Submodule R N) [hI : IsPrincipal (S.comap f)] : IsPrincipal S :=

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

@@ -419,7 +419,7 @@ theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u2}} [_inst_4 : CommRing.{u1} R] [_inst_5 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_4))] [_inst_6 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_4)] [_inst_7 : Semiring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) (s : Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) (a : R), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4)))))))))) -> (forall (b : R), (Membership.Mem.{u1, u1} R (Multiset.{u1} R) (Multiset.hasMem.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_4 _inst_5 _inst_6 a)) -> (Membership.Mem.{u2, u2} S (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) (SetLike.hasMem.{u2, u2} (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) S (Submonoid.setLike.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7))))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) f b) s)) -> (forall (c : Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4))), Membership.Mem.{u2, u2} S (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) (SetLike.hasMem.{u2, u2} (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) S (Submonoid.setLike.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7))))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) f ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4))) R (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4))) R (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4))) R (coeBase.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4))) R (Units.hasCoe.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4)))))) c)) s) -> (Membership.Mem.{u2, u2} S (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) (SetLike.hasMem.{u2, u2} (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) S (Submonoid.setLike.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7))))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) f a) s)
 but is expected to have type
-  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_4 : CommRing.{u2} R] [_inst_5 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))] [_inst_6 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_4)] [_inst_7 : Semiring.{u1} S] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (s : Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (a : R), (Ne.{succ u2} R a (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4) _inst_5)))))) -> (forall (b : R), (Membership.mem.{u2, u2} R (Multiset.{u2} R) (Multiset.instMembershipMultiset.{u2} R) b (PrincipalIdealRing.factors.{u2} R _inst_4 _inst_5 _inst_6 a)) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) b) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f b) s)) -> (forall (c : Units.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))), Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) c)) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) c)) s) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) a) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f a) s)
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_4 : CommRing.{u2} R] [_inst_5 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))] [_inst_6 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_4)] [_inst_7 : Semiring.{u1} S] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (s : Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (a : R), (Ne.{succ u2} R a (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4) _inst_5)))))) -> (forall (b : R), (Membership.mem.{u2, u2} R (Multiset.{u2} R) (Multiset.instMembershipMultiset.{u2} R) b (PrincipalIdealRing.factors.{u2} R _inst_4 _inst_5 _inst_6 a)) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) b) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f b) s)) -> (forall (c : Units.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))), Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R 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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) c)) s) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) a) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f a) s)
 Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.ring_hom_mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.ringHom_mem_submonoid_of_factors_subset_of_units_subsetₓ'. -/
 /-- If a `ring_hom` maps all units and all factors of an element `a` into a submonoid `s`, then it
 also maps `a` into that submonoid. -/
@@ -467,7 +467,7 @@ theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjec
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_4 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))), (Function.Surjective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) f)) -> (forall (I : Ideal.{u2} S (Ring.toSemiring.{u2} S _inst_4)) [hI : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S _inst_4) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) f I)], Submodule.IsPrincipal.{u2, u2} S S _inst_4 (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) (Semiring.toModule.{u2} S (Ring.toSemiring.{u2} S _inst_4)) I)
 but is expected to have type
-  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_4 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))), (Function.Surjective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))))) f)) -> (forall (I : Ideal.{u1} S (Ring.toSemiring.{u1} S _inst_4)) [hI : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Ideal.comap.{u2, u1, max u2 u1} R S (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S _inst_4) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) f I)], Submodule.IsPrincipal.{u1, u1} S S _inst_4 (Ring.toAddCommGroup.{u1} S _inst_4) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S _inst_4)) I)
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_4 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))), (Function.Surjective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))))) f)) -> (forall (I : Ideal.{u1} S (Ring.toSemiring.{u1} S _inst_4)) [hI : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Ideal.comap.{u2, u1, max u2 u1} R S (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S _inst_4) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) f I)], Submodule.IsPrincipal.{u1, u1} S S _inst_4 (Ring.toAddCommGroup.{u1} S _inst_4) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S _inst_4)) I)
 Case conversion may be inaccurate. Consider using '#align ideal.is_principal.of_comap Ideal.IsPrincipal.of_comapₓ'. -/
 theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f) (I : Ideal S)
     [hI : IsPrincipal (I.comap f)] : IsPrincipal I :=
@@ -480,7 +480,7 @@ theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f)
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_4 : Ring.{u2} S] [_inst_7 : IsPrincipalIdealRing.{u1} R _inst_1] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))), (Function.Surjective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) f)) -> (IsPrincipalIdealRing.{u2} S _inst_4)
 but is expected to have type
-  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_4 : Ring.{u1} S] [_inst_7 : IsPrincipalIdealRing.{u2} R _inst_1] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))), (Function.Surjective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))))) f)) -> (IsPrincipalIdealRing.{u1} S _inst_4)
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_4 : Ring.{u1} S] [_inst_7 : IsPrincipalIdealRing.{u2} R _inst_1] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))), (Function.Surjective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))))) f)) -> (IsPrincipalIdealRing.{u1} S _inst_4)
 Case conversion may be inaccurate. Consider using '#align is_principal_ideal_ring.of_surjective IsPrincipalIdealRing.of_surjectiveₓ'. -/
 /-- The surjective image of a principal ideal ring is again a principal ideal ring. -/
 theorem IsPrincipalIdealRing.of_surjective [IsPrincipalIdealRing R] (f : R →+* S)

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

@@ -185,7 +185,7 @@ theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_p
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N) _inst_4) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) => M -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ x))
 but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ x))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : M) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ x))
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
@@ -199,7 +199,7 @@ theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.ma
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N) _inst_4) (coeFn.{max 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R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) => (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
 but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) (fun (_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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : 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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R 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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ (Subtype.mk.{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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) (fun (_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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : 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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R 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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ (Subtype.mk.{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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_submodule_image_dvd_of_mem Submodule.IsPrincipal.generator_submoduleImage_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
@@ -454,7 +454,7 @@ variable [Module R M] [Module R N]
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} N] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_6 : Module.{u1, u3} R N (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3)] (f : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u2, succ u3} M N (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) (fun (_x : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) => M -> N) (LinearMap.hasCoeToFun.{u1, u1, u2, u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) -> (forall (S : Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u1, u2} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u1, u1, u2, u3, max u2 u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f S)], Submodule.IsPrincipal.{u1, u3} R N _inst_1 _inst_3 _inst_6 S)
 but is expected to have type
-  forall {R : Type.{u2}} {M : Type.{u3}} {N : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u1} N] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_6 : Module.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3)] (f : LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u3, succ u1} M N (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => N) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) -> (forall (S : Submodule.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u2, u3} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u2, u2, u3, u1, max u3 u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f S)], Submodule.IsPrincipal.{u2, u1} R N _inst_1 _inst_3 _inst_6 S)
+  forall {R : Type.{u2}} {M : Type.{u3}} {N : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u1} N] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_6 : Module.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3)] (f : LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u3, succ u1} M N (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6191 : M) => N) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) -> (forall (S : Submodule.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u2, u3} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u2, u2, u3, u1, max u3 u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f S)], Submodule.IsPrincipal.{u2, u1} R N _inst_1 _inst_3 _inst_6 S)
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.of_comap Submodule.IsPrincipal.of_comapₓ'. -/
 theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjective f)
     (S : Submodule R N) [hI : IsPrincipal (S.comap f)] : IsPrincipal S :=

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

@@ -197,7 +197,7 @@ theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.ma
 
 /- warning: submodule.is_principal.generator_submodule_image_dvd_of_mem -> Submodule.IsPrincipal.generator_submoduleImage_dvd_of_mem is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) 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(CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N) _inst_4) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) => (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
 but is expected to have type
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) (fun (_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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : 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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R 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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ (Subtype.mk.{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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_submodule_image_dvd_of_mem Submodule.IsPrincipal.generator_submoduleImage_dvd_of_memₓ'. -/
@@ -716,7 +716,7 @@ theorem nonPrincipals_eq_empty_iff : nonPrincipals R = ∅ ↔ IsPrincipalIdealR
 
 /- warning: non_principals_zorn -> nonPrincipals_zorn is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSubset.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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)))))))) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => Exists.{0} (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (fun (H : Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) => forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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))))))) J I)))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSubset.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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)))))))) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => Exists.{0} (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (fun (H : Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) => forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toHasLe.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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))))))) J I)))))
 but is expected to have type
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4546 : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4548 : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4546 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4548) c) -> (forall {K : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
 Case conversion may be inaccurate. Consider using '#align non_principals_zorn nonPrincipals_zornₓ'. -/

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

@@ -718,7 +718,7 @@ theorem nonPrincipals_eq_empty_iff : nonPrincipals R = ∅ ↔ IsPrincipalIdealR
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSubset.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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)))))))) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => Exists.{0} (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (fun (H : Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) => forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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))))))) J I)))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4647 : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4649 : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4647 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4649) c) -> (forall {K : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4546 : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4548 : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4546 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4548) c) -> (forall {K : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
 Case conversion may be inaccurate. Consider using '#align non_principals_zorn nonPrincipals_zornₓ'. -/
 /-- Any chain in the set of non-principal ideals has an upper bound which is non-principal.
 (Namely, the union of the chain is such an upper bound.)

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

@@ -727,11 +727,11 @@ theorem nonPrincipals_zorn (c : Set (Ideal R)) (hs : c ⊆ nonPrincipals R)
     (hchain : IsChain (· ≤ ·) c) {K : Ideal R} (hKmem : K ∈ c) :
     ∃ I ∈ nonPrincipals R, ∀ J ∈ c, J ≤ I :=
   by
-  refine' ⟨Sup c, _, fun J hJ => le_supₛ hJ⟩
+  refine' ⟨Sup c, _, fun J hJ => le_sSup hJ⟩
   rintro ⟨x, hx⟩
   have hxmem : x ∈ Sup c := hx.symm ▸ Submodule.mem_span_singleton_self x
-  obtain ⟨J, hJc, hxJ⟩ := (Submodule.mem_supₛ_of_directed ⟨K, hKmem⟩ hchain.directed_on).1 hxmem
-  have hSupJ : Sup c = J := le_antisymm (by simp [hx, Ideal.span_le, hxJ]) (le_supₛ hJc)
+  obtain ⟨J, hJc, hxJ⟩ := (Submodule.mem_sSup_of_directed ⟨K, hKmem⟩ hchain.directed_on).1 hxmem
+  have hSupJ : Sup c = J := le_antisymm (by simp [hx, Ideal.span_le, hxJ]) (le_sSup hJc)
   specialize hs hJc
   rw [← hSupJ, hx, nonPrincipals_def] at hs
   exact hs ⟨⟨x, rfl⟩⟩

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

@@ -172,7 +172,7 @@ theorem mem_iff_generator_dvd (S : Ideal R) [S.IsPrincipal] {x : R} : x ∈ S 
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_2 : CommRing.{u1} R] (S : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S] [is_prime : Ideal.IsPrime.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) S], (Ne.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S (Bot.bot.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.hasBot.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))))) -> (Prime.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S _inst_4))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_2 : CommRing.{u1} R] (S : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S] [is_prime : Ideal.IsPrime.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) S], (Ne.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S (Bot.bot.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.instBotSubmodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))))) -> (Prime.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S _inst_4))
+  forall {R : Type.{u1}} [_inst_2 : CommRing.{u1} R] (S : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) S] [is_prime : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) S], (Ne.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) S (Bot.bot.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.instBotSubmodule.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))))) -> (Prime.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) S _inst_4))
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.prime_generator_of_is_prime Submodule.IsPrincipal.prime_generator_of_isPrimeₓ'. -/
 theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_prime : S.IsPrime]
     (ne_bot : S ≠ ⊥) : Prime (generator S) :=
@@ -185,7 +185,7 @@ theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_p
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N) _inst_4) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) => M -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ x))
 but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ x))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R M R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ x))
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
@@ -199,7 +199,7 @@ theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.ma
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N) _inst_4) (coeFn.{max 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R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) => (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
 but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) (fun (_x : Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) O ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) (fun (_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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : 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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R 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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))))) ϕ (Subtype.mk.{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_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_submodule_image_dvd_of_mem Submodule.IsPrincipal.generator_submoduleImage_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
@@ -222,7 +222,7 @@ open Submodule.IsPrincipal Ideal
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {S : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} [hpi : Ideal.IsPrime.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) S], (Ne.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) S (Bot.bot.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasBot.{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.IsMaximal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) S)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {S : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} [hpi : Ideal.IsPrime.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) S], (Ne.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) S (Bot.bot.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.instBotSubmodule.{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.IsMaximal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) S)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {S : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))} [hpi : Ideal.IsPrime.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) S], (Ne.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) S (Bot.bot.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.instBotSubmodule.{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)))))) -> (Ideal.IsMaximal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) S)
 Case conversion may be inaccurate. Consider using '#align is_prime.to_maximal_ideal IsPrime.to_maximal_idealₓ'. -/
 -- TODO -- for a non-ID one could perhaps prove that if p < q are prime then q maximal;
 -- 0 isn't prime in a non-ID PIR but the Krull dimension is still <= 1.
@@ -330,7 +330,7 @@ instance (priority := 100) isNoetherianRing [Ring R] [IsPrincipalIdealRing R] :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) -> (Ideal.IsMaximal.{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) p)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) -> (Ideal.IsMaximal.{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.instSingletonSet.{u1} R) p)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) p) -> (Ideal.IsMaximal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{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) p)))
 Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.is_maximal_of_irreducible PrincipalIdealRing.isMaximal_of_irreducibleₓ'. -/
 theorem isMaximal_of_irreducible [CommRing R] [IsPrincipalIdealRing R] {p : R}
     (hp : Irreducible p) : Ideal.IsMaximal (span R ({p} : Set R)) :=
@@ -349,7 +349,7 @@ variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, Iff (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) (Prime.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) p)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, Iff (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) (Prime.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)) p)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, Iff (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) p) (Prime.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)) p)
 Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.irreducible_iff_prime PrincipalIdealRing.irreducible_iff_primeₓ'. -/
 theorem irreducible_iff_prime {p : R} : Irreducible p ↔ Prime p :=
   ⟨fun hp => (Ideal.span_singleton_prime hp.NeZero).1 <| (isMaximal_of_irreducible hp).IsPrime,
@@ -360,7 +360,7 @@ theorem irreducible_iff_prime {p : R} : Irreducible p ↔ Prime p :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : Associates.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (Irreducible.{u1} (Associates.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) (MonoidWithZero.toMonoid.{u1} (Associates.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommMonoidWithZero.toMonoidWithZero.{u1} (Associates.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Associates.commMonoidWithZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) p) (Prime.{u1} (Associates.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Associates.commMonoidWithZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))}, Iff (Irreducible.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (MonoidWithZero.toMonoid.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (CommMonoidWithZero.toMonoidWithZero.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))) p) (Prime.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) p)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))}, Iff (Irreducible.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (MonoidWithZero.toMonoid.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (CommMonoidWithZero.toMonoidWithZero.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))) p) (Prime.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) p)
 Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.associates_irreducible_iff_prime PrincipalIdealRing.associates_irreducible_iff_primeₓ'. -/
 theorem associates_irreducible_iff_prime : ∀ {p : Associates R}, Irreducible p ↔ Prime p :=
   Associates.irreducible_iff_prime_iff.1 fun _ => irreducible_iff_prime
@@ -381,7 +381,7 @@ noncomputable def factors (a : R) : Multiset R :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] (a : R), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) -> (And (forall (b : R), (Membership.Mem.{u1, u1} R (Multiset.{u1} R) (Multiset.hasMem.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) -> (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) b)) (Associated.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) a))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] (a : R), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) -> (And (forall (b : R), (Membership.mem.{u1, u1} R (Multiset.{u1} R) (Multiset.instMembershipMultiset.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) -> (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) b)) (Associated.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Multiset.prod.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) a))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] (a : R), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) -> (And (forall (b : R), (Membership.mem.{u1, u1} R (Multiset.{u1} R) (Multiset.instMembershipMultiset.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) -> (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) b)) (Associated.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Multiset.prod.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) a))
 Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.factors_spec PrincipalIdealRing.factors_specₓ'. -/
 theorem factors_spec (a : R) (h : a ≠ 0) :
     (∀ b ∈ factors a, Irreducible b) ∧ Associated (factors a).Prod a :=
@@ -394,7 +394,7 @@ theorem factors_spec (a : R) (h : a ≠ 0) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_4 : CommRing.{u1} R] [_inst_5 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_4))] [_inst_6 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_4)] {a : R} {b : R}, (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4)))))))))) -> (Membership.Mem.{u1, u1} R (Multiset.{u1} R) (Multiset.hasMem.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_4 _inst_5 _inst_6 a)) -> (Ne.{succ u1} R b (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))))))))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_4 : CommRing.{u1} R] [_inst_5 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_4))] [_inst_6 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_4)] {a : R} {b : R}, (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_4) _inst_5)))))) -> (Membership.mem.{u1, u1} R (Multiset.{u1} R) (Multiset.instMembershipMultiset.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_4 _inst_5 _inst_6 a)) -> (Ne.{succ u1} R b (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_4) _inst_5))))))
+  forall {R : Type.{u1}} [_inst_4 : CommRing.{u1} R] [_inst_5 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_4))] [_inst_6 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_4)] {a : R} {b : R}, (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_4) _inst_5)))))) -> (Membership.mem.{u1, u1} R (Multiset.{u1} R) (Multiset.instMembershipMultiset.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_4 _inst_5 _inst_6 a)) -> (Ne.{succ u1} R b (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_4) _inst_5))))))
 Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.ne_zero_of_mem_factors PrincipalIdealRing.ne_zero_of_mem_factorsₓ'. -/
 theorem ne_zero_of_mem_factors {R : Type v} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
     {a b : R} (ha : a ≠ 0) (hb : b ∈ factors a) : b ≠ 0 :=
@@ -405,7 +405,7 @@ theorem ne_zero_of_mem_factors {R : Type v} [CommRing R] [IsDomain R] [IsPrincip
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] (s : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) {a : R}, (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) -> (forall (b : R), (Membership.Mem.{u1, u1} R (Multiset.{u1} R) (Multiset.hasMem.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) -> (Membership.Mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (SetLike.hasMem.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) b s)) -> (forall (c : Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))), Membership.Mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (SetLike.hasMem.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (coeBase.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Units.hasCoe.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) c) s) -> (Membership.Mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (SetLike.hasMem.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) R (Submonoid.setLike.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) a s)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] (s : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) {a : R}, (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) -> (forall (b : R), (Membership.mem.{u1, u1} R (Multiset.{u1} R) (Multiset.instMembershipMultiset.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) -> (Membership.mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) b s)) -> (forall (c : Units.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))), Membership.mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (Units.val.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c) s) -> (Membership.mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) a s)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] (s : Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) {a : R}, (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) -> (forall (b : R), (Membership.mem.{u1, u1} R (Multiset.{u1} R) (Multiset.instMembershipMultiset.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) -> (Membership.mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))))) b s)) -> (forall (c : Units.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))), Membership.mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Units.val.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) c) s) -> (Membership.mem.{u1, u1} R (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))))) a s)
 Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.mem_submonoid_of_factors_subset_of_units_subsetₓ'. -/
 theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R} (ha : a ≠ 0)
     (hfac : ∀ b ∈ factors a, b ∈ s) (hunit : ∀ c : Rˣ, (c : R) ∈ s) : a ∈ s :=
@@ -419,7 +419,7 @@ theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u2}} [_inst_4 : CommRing.{u1} R] [_inst_5 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_4))] [_inst_6 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_4)] [_inst_7 : Semiring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) (s : Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) (a : R), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4)))))))))) -> (forall (b : R), (Membership.Mem.{u1, u1} R (Multiset.{u1} R) (Multiset.hasMem.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_4 _inst_5 _inst_6 a)) -> (Membership.Mem.{u2, u2} S (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) (SetLike.hasMem.{u2, u2} (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) S (Submonoid.setLike.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7))))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) f b) s)) -> (forall (c : Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4))), Membership.Mem.{u2, u2} S (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) (SetLike.hasMem.{u2, u2} (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) S (Submonoid.setLike.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7))))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) f ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4))) R (HasLiftT.mk.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4))) R (CoeTCₓ.coe.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4))) R (coeBase.{succ u1, succ u1} (Units.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4))) R (Units.hasCoe.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_4)))))) c)) s) -> (Membership.Mem.{u2, u2} S (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) (SetLike.hasMem.{u2, u2} (Submonoid.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7)))) S (Submonoid.setLike.{u2} S (MulZeroOneClass.toMulOneClass.{u2} S (NonAssocSemiring.toMulZeroOneClass.{u2} S (Semiring.toNonAssocSemiring.{u2} S _inst_7))))) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_4))) (Semiring.toNonAssocSemiring.{u2} S _inst_7)) f a) s)
 but is expected to have type
-  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_4 : CommRing.{u2} R] [_inst_5 : IsDomain.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_4))] [_inst_6 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_4)] [_inst_7 : Semiring.{u1} S] (f : RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (s : Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (a : R), (Ne.{succ u2} R a (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4) _inst_5)))))) -> (forall (b : R), (Membership.mem.{u2, u2} R (Multiset.{u2} R) (Multiset.instMembershipMultiset.{u2} R) b (PrincipalIdealRing.factors.{u2} R _inst_4 _inst_5 _inst_6 a)) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) b) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f b) s)) -> (forall (c : Units.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_4))))), Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_4)))) c)) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_4)))) c)) s) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) a) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f a) s)
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_4 : CommRing.{u2} R] [_inst_5 : IsDomain.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))] [_inst_6 : IsPrincipalIdealRing.{u2} R (CommRing.toRing.{u2} R _inst_4)] [_inst_7 : Semiring.{u1} S] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (s : Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (a : R), (Ne.{succ u2} R a (OfNat.ofNat.{u2} R 0 (Zero.toOfNat0.{u2} R (CommMonoidWithZero.toZero.{u2} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u2} R (IsDomain.toCancelCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4) _inst_5)))))) -> (forall (b : R), (Membership.mem.{u2, u2} R (Multiset.{u2} R) (Multiset.instMembershipMultiset.{u2} R) b (PrincipalIdealRing.factors.{u2} R _inst_4 _inst_5 _inst_6 a)) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) b) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f b) s)) -> (forall (c : Units.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))), Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) c)) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) c)) s) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) a) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f a) s)
 Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.ring_hom_mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.ringHom_mem_submonoid_of_factors_subset_of_units_subsetₓ'. -/
 /-- If a `ring_hom` maps all units and all factors of an element `a` into a submonoid `s`, then it
 also maps `a` into that submonoid. -/
@@ -454,7 +454,7 @@ variable [Module R M] [Module R N]
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} N] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_6 : Module.{u1, u3} R N (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3)] (f : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u2, succ u3} M N (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) (fun (_x : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) => M -> N) (LinearMap.hasCoeToFun.{u1, u1, u2, u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) -> (forall (S : Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u1, u2} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u1, u1, u2, u3, max u2 u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f S)], Submodule.IsPrincipal.{u1, u3} R N _inst_1 _inst_3 _inst_6 S)
 but is expected to have type
-  forall {R : Type.{u2}} {M : Type.{u3}} {N : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u1} N] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_6 : Module.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3)] (f : LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u3, succ u1} M N (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => N) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) f)) -> (forall (S : Submodule.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u2, u3} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u2, u2, u3, u1, max u3 u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) f S)], Submodule.IsPrincipal.{u2, u1} R N _inst_1 _inst_3 _inst_6 S)
+  forall {R : Type.{u2}} {M : Type.{u3}} {N : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u1} N] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_6 : Module.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3)] (f : LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u3, succ u1} M N (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => N) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f)) -> (forall (S : Submodule.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u2, u3} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u2, u2, u3, u1, max u3 u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) f S)], Submodule.IsPrincipal.{u2, u1} R N _inst_1 _inst_3 _inst_6 S)
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.of_comap Submodule.IsPrincipal.of_comapₓ'. -/
 theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjective f)
     (S : Submodule R N) [hI : IsPrincipal (S.comap f)] : IsPrincipal S :=
@@ -467,7 +467,7 @@ theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjec
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_4 : Ring.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))), (Function.Surjective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) f)) -> (forall (I : Ideal.{u2} S (Ring.toSemiring.{u2} S _inst_4)) [hI : Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S _inst_4) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) f I)], Submodule.IsPrincipal.{u2, u2} S S _inst_4 (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) (Semiring.toModule.{u2} S (Ring.toSemiring.{u2} S _inst_4)) I)
 but is expected to have type
-  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_4 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))), (Function.Surjective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4)))))) f)) -> (forall (I : Ideal.{u1} S (Ring.toSemiring.{u1} S _inst_4)) [hI : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Ideal.comap.{u2, u1, max u2 u1} R S (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S _inst_4) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) f I)], Submodule.IsPrincipal.{u1, u1} S S _inst_4 (Ring.toAddCommGroup.{u1} S _inst_4) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S _inst_4)) I)
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_4 : Ring.{u1} S] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))), (Function.Surjective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))))) f)) -> (forall (I : Ideal.{u1} S (Ring.toSemiring.{u1} S _inst_4)) [hI : Submodule.IsPrincipal.{u2, u2} R R _inst_1 (Ring.toAddCommGroup.{u2} R _inst_1) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Ideal.comap.{u2, u1, max u2 u1} R S (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S _inst_4) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) f I)], Submodule.IsPrincipal.{u1, u1} S S _inst_4 (Ring.toAddCommGroup.{u1} S _inst_4) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S _inst_4)) I)
 Case conversion may be inaccurate. Consider using '#align ideal.is_principal.of_comap Ideal.IsPrincipal.of_comapₓ'. -/
 theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f) (I : Ideal S)
     [hI : IsPrincipal (I.comap f)] : IsPrincipal I :=
@@ -480,7 +480,7 @@ theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f)
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_4 : Ring.{u2} S] [_inst_7 : IsPrincipalIdealRing.{u1} R _inst_1] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))), (Function.Surjective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) f)) -> (IsPrincipalIdealRing.{u2} S _inst_4)
 but is expected to have type
-  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_4 : Ring.{u1} S] [_inst_7 : IsPrincipalIdealRing.{u2} R _inst_1] (f : RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))), (Function.Surjective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4))) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u1} S (Ring.toNonAssocRing.{u1} S _inst_4)))))) f)) -> (IsPrincipalIdealRing.{u1} S _inst_4)
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_4 : Ring.{u1} S] [_inst_7 : IsPrincipalIdealRing.{u2} R _inst_1] (f : RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))), (Function.Surjective.{succ u2, succ u1} R S (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4))) R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S _inst_4)))))) f)) -> (IsPrincipalIdealRing.{u1} S _inst_4)
 Case conversion may be inaccurate. Consider using '#align is_principal_ideal_ring.of_surjective IsPrincipalIdealRing.of_surjectiveₓ'. -/
 /-- The surjective image of a principal ideal ring is again a principal ideal ring. -/
 theorem IsPrincipalIdealRing.of_surjective [IsPrincipalIdealRing R] (f : R →+* S)
@@ -524,7 +524,7 @@ theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (a : R) (b : R) {z : R}, Iff (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 a b) z) (Exists.{succ u1} R (fun (x : R) => Exists.{succ u1} R (fun (y : R) => Eq.{succ u1} R z (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (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 x) (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)))) b y)))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (a : R) (b : R) {z : R}, Iff (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 a b) z) (Exists.{succ u1} R (fun (x : R) => Exists.{succ u1} R (fun (y : R) => Eq.{succ u1} R z (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (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 x) (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))))) b y)))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (a : R) (b : R) {z : R}, Iff (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 a b) z) (Exists.{succ u1} R (fun (x : R) => Exists.{succ u1} R (fun (y : R) => Eq.{succ u1} R z (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (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 x) (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))))) b y)))))
 Case conversion may be inaccurate. Consider using '#align gcd_dvd_iff_exists gcd_dvd_iff_existsₓ'. -/
 theorem gcd_dvd_iff_exists (a b : R) {z} : gcd a b ∣ z ↔ ∃ x y, z = a * x + b * y := by
   simp_rw [mul_comm a, mul_comm b, @eq_comm _ z, ← Ideal.mem_span_pair, ← span_gcd,
@@ -535,7 +535,7 @@ theorem gcd_dvd_iff_exists (a b : R) {z} : gcd a b ∣ z ↔ ∃ x y, z = a * x
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (a : R) (b : R), Exists.{succ u1} R (fun (x : R) => Exists.{succ u1} R (fun (y : R) => Eq.{succ u1} R (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 a b) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (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 x) (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)))) b y))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (a : R) (b : R), Exists.{succ u1} R (fun (x : R) => Exists.{succ u1} R (fun (y : R) => Eq.{succ u1} R (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 a b) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (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 x) (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))))) b y))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (a : R) (b : R), Exists.{succ u1} R (fun (x : R) => Exists.{succ u1} R (fun (y : R) => Eq.{succ u1} R (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 a b) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) (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 x) (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))))) b y))))
 Case conversion may be inaccurate. Consider using '#align exists_gcd_eq_mul_add_mul exists_gcd_eq_mul_add_mulₓ'. -/
 /-- **Bézout's lemma** -/
 theorem exists_gcd_eq_mul_add_mul (a b : R) : ∃ x y, gcd a b = a * x + b * y := by
@@ -546,7 +546,7 @@ theorem exists_gcd_eq_mul_add_mul (a b : R) : ∃ x y, gcd a b = a * x + b * y :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), Iff (IsUnit.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 x y)) (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), Iff (IsUnit.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 x y)) (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), Iff (IsUnit.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 x y)) (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
 Case conversion may be inaccurate. Consider using '#align gcd_is_unit_iff gcd_isUnit_iffₓ'. -/
 theorem gcd_isUnit_iff (x y : R) : IsUnit (gcd x y) ↔ IsCoprime x y := by
   rw [IsCoprime, ← Ideal.mem_span_pair, ← span_gcd, ← span_singleton_eq_top, eq_top_iff_one]
@@ -556,7 +556,7 @@ theorem gcd_isUnit_iff (x y : R) : IsUnit (gcd x y) ↔ IsCoprime x y := by
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), (Not (And (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) (Eq.{succ u1} R y (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))))) -> (forall (z : R), (Membership.Mem.{u1, u1} R (Set.{u1} R) (Set.hasMem.{u1} R) z (nonunits.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) -> (Ne.{succ u1} R z (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) -> (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z x) -> (Not (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z y))) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), (Not (And (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) (Eq.{succ u1} R y (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))))) -> (forall (z : R), (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) z (nonunits.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) -> (Ne.{succ u1} R z (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z x) -> (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z y))) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), (Not (And (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) (Eq.{succ u1} R y (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))))) -> (forall (z : R), (Membership.mem.{u1, u1} R (Set.{u1} R) (Set.instMembershipSet.{u1} R) z (nonunits.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) -> (Ne.{succ u1} R z (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z x) -> (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z y))) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
 Case conversion may be inaccurate. Consider using '#align is_coprime_of_dvd isCoprime_of_dvdₓ'. -/
 -- this should be proved for UFDs surely?
 theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
@@ -572,7 +572,7 @@ theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) x) -> (Or (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) x y) (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) x) -> (Or (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) x y) (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) x) -> (Or (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) x y) (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y))
 Case conversion may be inaccurate. Consider using '#align dvd_or_coprime dvd_or_coprimeₓ'. -/
 -- this should be proved for UFDs surely?
 theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y :=
@@ -590,7 +590,7 @@ theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {x : R} {y : R}, (Not (And (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) (Eq.{succ u1} R y (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))))) -> (forall (z : R), (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) z) -> (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z x) -> (Not (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z y))) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {x : R} {y : R}, (Not (And (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) (Eq.{succ u1} R y (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))))) -> (forall (z : R), (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) z) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z x) -> (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z y))) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {x : R} {y : R}, (Not (And (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) (Eq.{succ u1} R y (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))))) -> (forall (z : R), (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) z) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z x) -> (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z y))) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
 Case conversion may be inaccurate. Consider using '#align is_coprime_of_irreducible_dvd isCoprime_of_irreducible_dvdₓ'. -/
 theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z : R, Irreducible z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
@@ -607,7 +607,7 @@ theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {x : R} {y : R}, (Not (And (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) (Eq.{succ u1} R y (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))))) -> (forall (z : R), (Prime.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) z) -> (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z x) -> (Not (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z y))) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {x : R} {y : R}, (Not (And (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) (Eq.{succ u1} R y (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))))) -> (forall (z : R), (Prime.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)) z) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z x) -> (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z y))) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {x : R} {y : R}, (Not (And (Eq.{succ u1} R x (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) (Eq.{succ u1} R y (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))))) -> (forall (z : R), (Prime.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)) z) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z x) -> (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) z y))) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
 Case conversion may be inaccurate. Consider using '#align is_coprime_of_prime_dvd isCoprime_of_prime_dvdₓ'. -/
 theorem isCoprime_of_prime_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z : R, Prime z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
@@ -618,7 +618,7 @@ theorem isCoprime_of_prime_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {n : R}, (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) -> (Iff (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p n) (Not (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p n)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {n : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) -> (Iff (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p n) (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p n)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {n : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) p) -> (Iff (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p n) (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p n)))
 Case conversion may be inaccurate. Consider using '#align irreducible.coprime_iff_not_dvd Irreducible.coprime_iff_not_dvdₓ'. -/
 theorem Irreducible.coprime_iff_not_dvd {p n : R} (pp : Irreducible p) : IsCoprime p n ↔ ¬p ∣ n :=
   by
@@ -647,7 +647,7 @@ theorem Prime.coprime_iff_not_dvd {p n : R} (pp : Prime p) : IsCoprime p n ↔ 
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {n : R}, (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) -> (Iff (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p n) (Not (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p n)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {n : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) -> (Iff (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p n) (Not (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p n)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {n : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) p) -> (Iff (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p n) (Not (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p n)))
 Case conversion may be inaccurate. Consider using '#align irreducible.dvd_iff_not_coprime Irreducible.dvd_iff_not_coprimeₓ'. -/
 theorem Irreducible.dvd_iff_not_coprime {p n : R} (hp : Irreducible p) : p ∣ n ↔ ¬IsCoprime p n :=
   iff_not_comm.2 hp.coprime_iff_not_dvd
@@ -657,7 +657,7 @@ theorem Irreducible.dvd_iff_not_coprime {p n : R} (hp : Irreducible p) : p ∣ n
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {a : R} (m : Nat), (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) -> (Not (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p a)) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) a (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p m))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {a : R} (m : Nat), (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) -> (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p a)) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) a (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 (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) p m))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {a : R} (m : Nat), (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) p) -> (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p a)) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) a (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 (CommRing.toCommSemiring.{u1} R _inst_1)))))) p m))
 Case conversion may be inaccurate. Consider using '#align irreducible.coprime_pow_of_not_dvd Irreducible.coprime_pow_of_not_dvdₓ'. -/
 theorem Irreducible.coprime_pow_of_not_dvd {p a : R} (m : ℕ) (hp : Irreducible p) (h : ¬p ∣ a) :
     IsCoprime a (p ^ m) :=
@@ -668,7 +668,7 @@ theorem Irreducible.coprime_pow_of_not_dvd {p a : R} (m : ℕ) (hp : Irreducible
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R}, (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) -> (forall (i : R), Or (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p i) (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p i))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) -> (forall (i : R), Or (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p i) (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p i))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) p) -> (forall (i : R), Or (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p i) (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalCommSemiring.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalCommSemiring.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p i))
 Case conversion may be inaccurate. Consider using '#align irreducible.coprime_or_dvd Irreducible.coprime_or_dvdₓ'. -/
 theorem Irreducible.coprime_or_dvd {p : R} (hp : Irreducible p) (i : R) : IsCoprime p i ∨ p ∣ i :=
   (em _).imp_right hp.dvd_iff_not_coprime.2
@@ -678,7 +678,7 @@ theorem Irreducible.coprime_or_dvd {p : R} (hp : Irreducible p) (i : R) : IsCopr
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {a : R} {b : R} {c : R}, (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) a b) -> (forall {k : Nat}, (Eq.{succ 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) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c k)) -> (Exists.{succ u1} R (fun (d : R) => Associated.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) d k) a)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {a : R} {b : R} {c : R}, (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) a b) -> (forall {k : Nat}, (Eq.{succ 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) (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 (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) c k)) -> (Exists.{succ u1} R (fun (d : R) => Associated.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (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 (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) d k) a)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {a : R} {b : R} {c : R}, (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) a b) -> (forall {k : Nat}, (Eq.{succ 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) (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 (CommRing.toCommSemiring.{u1} R _inst_1)))))) c k)) -> (Exists.{succ u1} R (fun (d : R) => Associated.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (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 (CommRing.toCommSemiring.{u1} R _inst_1)))))) d k) a)))
 Case conversion may be inaccurate. Consider using '#align exists_associated_pow_of_mul_eq_pow' exists_associated_pow_of_mul_eq_pow'ₓ'. -/
 theorem exists_associated_pow_of_mul_eq_pow' {a b c : R} (hab : IsCoprime a b) {k : ℕ}
     (h : a * b = c ^ k) : ∃ d, Associated (d ^ k) a :=
@@ -718,7 +718,7 @@ theorem nonPrincipals_eq_empty_iff : nonPrincipals R = ∅ ↔ IsPrincipalIdealR
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSubset.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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)))))))) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => Exists.{0} (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (fun (H : Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) => forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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))))))) J I)))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4647 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4649 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4647 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4649) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4647 : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4649 : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4647 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4649) c) -> (forall {K : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
 Case conversion may be inaccurate. Consider using '#align non_principals_zorn nonPrincipals_zornₓ'. -/
 /-- Any chain in the set of non-principal ideals has an upper bound which is non-principal.
 (Namely, the union of the chain is such an upper bound.)

mathlib commit https://github.com/leanprover-community/mathlib/commit/36b8aa61ea7c05727161f96a0532897bd72aedab

@@ -718,7 +718,7 @@ theorem nonPrincipals_eq_empty_iff : nonPrincipals R = ∅ ↔ IsPrincipalIdealR
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSubset.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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)))))))) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => Exists.{0} (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (fun (H : Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) => forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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))))))) J I)))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4651 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4653 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4651 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4653) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4647 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4649 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4647 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4649) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
 Case conversion may be inaccurate. Consider using '#align non_principals_zorn nonPrincipals_zornₓ'. -/
 /-- Any chain in the set of non-principal ideals has an upper bound which is non-principal.
 (Namely, the union of the chain is such an upper bound.)

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

@@ -75,7 +75,7 @@ instance bot_isPrincipal : (⊥ : Submodule R M).IsPrincipal :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R], Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Top.top.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Submodule.hasTop.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R], Submodule.IsPrincipal.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u1} R _inst_1) (Top.top.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u1} R _inst_1)) (Submodule.instTopSubmodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u1} R _inst_1)))
+  forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R], Submodule.IsPrincipal.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Top.top.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Submodule.instTopSubmodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))))
 Case conversion may be inaccurate. Consider using '#align top_is_principal top_isPrincipalₓ'. -/
 instance top_isPrincipal : (⊤ : Submodule R R).IsPrincipal :=
   ⟨⟨1, Ideal.span_singleton_one.symm⟩⟩
@@ -185,7 +185,7 @@ theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_p
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u2 u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.semilinearMapClass.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N) _inst_4) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (fun (_x : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) => M -> R) (LinearMap.hasCoeToFun.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ x))
 but is expected to have type
-  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2))) _inst_3 (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ x))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N)] {x : M}, (Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N) -> (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.map.{u1, u1, u2, u1, max u1 u2} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (RingHomSurjective.ids.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (LinearMap.instSemilinearMapClassLinearMap.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M R (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R M R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) _inst_3 (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ x))
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
@@ -199,7 +199,7 @@ theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.ma
 lean 3 declaration is
   forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (coeSort.{succ u2, succ (succ u2)} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) O) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N) _inst_4) (coeFn.{max 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 but is expected to have type
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(AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R 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(_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : CommRing.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] {N : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} {O : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3} (hNO : LE.le.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.completeLattice.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))))) N O) (ϕ : LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N)] {x : M} (hx : Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x N), Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_2)))))) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (LinearMap.submoduleImage.{u1, u2, u1} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) O ϕ N) _inst_4) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (LinearMap.{u1, u1, u2, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) (fun (_x : Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
 Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_submodule_image_dvd_of_mem Submodule.IsPrincipal.generator_submoduleImage_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
@@ -718,7 +718,7 @@ theorem nonPrincipals_eq_empty_iff : nonPrincipals R = ∅ ↔ IsPrincipalIdealR
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSubset.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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)))))))) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => Exists.{0} (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (fun (H : Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) => forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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))))))) J I)))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4580 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4582 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4580 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4582) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4651 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4653 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4651 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4653) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
 Case conversion may be inaccurate. Consider using '#align non_principals_zorn nonPrincipals_zornₓ'. -/
 /-- Any chain in the set of non-principal ideals has an upper bound which is non-principal.
 (Namely, the union of the chain is such an upper bound.)

mathlib commit https://github.com/leanprover-community/mathlib/commit/09079525fd01b3dda35e96adaa08d2f943e1648c

@@ -53,7 +53,7 @@ section
 variable [Ring R] [AddCommGroup M] [Module R M]
 
 #print Submodule.IsPrincipal /-
-/- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
 /-- An `R`-submodule of `M` is principal if it is generated by one element. -/
 @[mk_iff]
 class Submodule.IsPrincipal (S : Submodule R M) : Prop where

mathlib commit https://github.com/leanprover-community/mathlib/commit/57e09a1296bfb4330ddf6624f1028ba186117d82

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Morenikeji Neri
 
 ! This file was ported from Lean 3 source module ring_theory.principal_ideal_domain
-! leanprover-community/mathlib commit 6010cf523816335f7bae7f8584cb2edaace73940
+! leanprover-community/mathlib commit c085f3044fe585c575e322bfab45b3633c48d820
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -14,6 +14,9 @@ import Mathbin.RingTheory.UniqueFactorizationDomain
 /-!
 # Principal ideal rings and principal ideal domains
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 A principal ideal ring (PIR) is a ring in which all left ideals are principal. A
 principal ideal domain (PID) is an integral domain which is a principal ideal ring.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/290a7ba01fbcab1b64757bdaa270d28f4dcede35

@@ -49,35 +49,53 @@ section
 
 variable [Ring R] [AddCommGroup M] [Module R M]
 
+#print Submodule.IsPrincipal /-
 /- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`principal] [] -/
 /-- An `R`-submodule of `M` is principal if it is generated by one element. -/
 @[mk_iff]
 class Submodule.IsPrincipal (S : Submodule R M) : Prop where
   principal : ∃ a, S = span R {a}
 #align submodule.is_principal Submodule.IsPrincipal
+-/
 
+/- warning: bot_is_principal -> bot_isPrincipal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)], Submodule.IsPrincipal.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))
+but is expected to have type
+  forall {R : Type.{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)], Submodule.IsPrincipal.{u1, u2} R M _inst_1 _inst_2 _inst_3 (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3) (Submodule.instBotSubmodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))
+Case conversion may be inaccurate. Consider using '#align bot_is_principal bot_isPrincipalₓ'. -/
 instance bot_isPrincipal : (⊥ : Submodule R M).IsPrincipal :=
   ⟨⟨0, by simp⟩⟩
 #align bot_is_principal bot_isPrincipal
 
+/- warning: top_is_principal -> top_isPrincipal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R], Submodule.IsPrincipal.{u1, u1} R R _inst_1 (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (Top.top.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (Submodule.hasTop.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R _inst_1))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : Ring.{u1} R], Submodule.IsPrincipal.{u1, u1} R R _inst_1 (Ring.toAddCommGroup.{u1} R _inst_1) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u1} R _inst_1) (Top.top.{u1} (Submodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u1} R _inst_1)) (Submodule.instTopSubmodule.{u1, u1} R R (Ring.toSemiring.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)))) (instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRing.{u1} R _inst_1)))
+Case conversion may be inaccurate. Consider using '#align top_is_principal top_isPrincipalₓ'. -/
 instance top_isPrincipal : (⊤ : Submodule R R).IsPrincipal :=
   ⟨⟨1, Ideal.span_singleton_one.symm⟩⟩
 #align top_is_principal top_isPrincipal
 
 variable (R)
 
+#print IsPrincipalIdealRing /-
 /-- A ring is a principal ideal ring if all (left) ideals are principal. -/
 @[mk_iff]
 class IsPrincipalIdealRing (R : Type u) [Ring R] : Prop where
   principal : ∀ S : Ideal R, S.IsPrincipal
 #align is_principal_ideal_ring IsPrincipalIdealRing
+-/
 
 attribute [instance] IsPrincipalIdealRing.principal
 
+#print DivisionRing.isPrincipalIdealRing /-
 instance (priority := 100) DivisionRing.isPrincipalIdealRing (K : Type u) [DivisionRing K] :
     IsPrincipalIdealRing K
     where principal S := by rcases Ideal.eq_bot_or_top S with (rfl | rfl) <;> infer_instance
 #align division_ring.is_principal_ideal_ring DivisionRing.isPrincipalIdealRing
+-/
 
 end
 
@@ -89,32 +107,48 @@ section Ring
 
 variable [Ring R] [Module R M]
 
+#print Submodule.IsPrincipal.generator /-
 /-- `generator I`, if `I` is a principal submodule, is an `x ∈ M` such that `span R {x} = I` -/
 noncomputable def generator (S : Submodule R M) [S.IsPrincipal] : M :=
   Classical.choose (principal S)
 #align submodule.is_principal.generator Submodule.IsPrincipal.generator
+-/
 
+#print Submodule.IsPrincipal.span_singleton_generator /-
 theorem span_singleton_generator (S : Submodule R M) [S.IsPrincipal] : span R {generator S} = S :=
   Eq.symm (Classical.choose_spec (principal S))
 #align submodule.is_principal.span_singleton_generator Submodule.IsPrincipal.span_singleton_generator
+-/
 
+#print Ideal.span_singleton_generator /-
 theorem Ideal.span_singleton_generator (I : Ideal R) [I.IsPrincipal] :
     Ideal.span ({generator I} : Set R) = I :=
   Eq.symm (Classical.choose_spec (principal I))
 #align ideal.span_singleton_generator Ideal.span_singleton_generator
+-/
 
+#print Submodule.IsPrincipal.generator_mem /-
 @[simp]
 theorem generator_mem (S : Submodule R M) [S.IsPrincipal] : generator S ∈ S :=
   by
   conv_rhs => rw [← span_singleton_generator S]
   exact subset_span (mem_singleton _)
 #align submodule.is_principal.generator_mem Submodule.IsPrincipal.generator_mem
+-/
 
+#print Submodule.IsPrincipal.mem_iff_eq_smul_generator /-
 theorem mem_iff_eq_smul_generator (S : Submodule R M) [S.IsPrincipal] {x : M} :
     x ∈ S ↔ ∃ s : R, x = s • generator S := by
   simp_rw [@eq_comm _ x, ← mem_span_singleton, span_singleton_generator]
 #align submodule.is_principal.mem_iff_eq_smul_generator Submodule.IsPrincipal.mem_iff_eq_smul_generator
+-/
 
+/- warning: submodule.is_principal.eq_bot_iff_generator_eq_zero -> Submodule.IsPrincipal.eq_bot_iff_generator_eq_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : Ring.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) [_inst_4 : Submodule.IsPrincipal.{u1, u2} R M _inst_2 _inst_1 _inst_3 S], Iff (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) S (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.hasBot.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))) (Eq.{succ u2} M (Submodule.IsPrincipal.generator.{u1, u2} R M _inst_1 _inst_2 _inst_3 S _inst_4) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (SubNegMonoid.toAddMonoid.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_1)))))))))
+but is expected to have type
+  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommGroup.{u2} M] [_inst_2 : Ring.{u1} R] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1)] (S : Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) [_inst_4 : Submodule.IsPrincipal.{u1, u2} R M _inst_2 _inst_1 _inst_3 S], Iff (Eq.{succ u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) S (Bot.bot.{u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (Submodule.instBotSubmodule.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_2) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3))) (Eq.{succ u2} M (Submodule.IsPrincipal.generator.{u1, u2} R M _inst_1 _inst_2 _inst_3 S _inst_4) (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_1))))))))
+Case conversion may be inaccurate. Consider using '#align submodule.is_principal.eq_bot_iff_generator_eq_zero Submodule.IsPrincipal.eq_bot_iff_generator_eq_zeroₓ'. -/
 theorem eq_bot_iff_generator_eq_zero (S : Submodule R M) [S.IsPrincipal] :
     S = ⊥ ↔ generator S = 0 := by rw [← @span_singleton_eq_bot R M, span_singleton_generator]
 #align submodule.is_principal.eq_bot_iff_generator_eq_zero Submodule.IsPrincipal.eq_bot_iff_generator_eq_zero
@@ -125,10 +159,18 @@ section CommRing
 
 variable [CommRing R] [Module R M]
 
+#print Submodule.IsPrincipal.mem_iff_generator_dvd /-
 theorem mem_iff_generator_dvd (S : Ideal R) [S.IsPrincipal] {x : R} : x ∈ S ↔ generator S ∣ x :=
   (mem_iff_eq_smul_generator S).trans (exists_congr fun a => by simp only [mul_comm, smul_eq_mul])
 #align submodule.is_principal.mem_iff_generator_dvd Submodule.IsPrincipal.mem_iff_generator_dvd
+-/
 
+/- warning: submodule.is_principal.prime_generator_of_is_prime -> Submodule.IsPrincipal.prime_generator_of_isPrime is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_2 : CommRing.{u1} R] (S : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S] [is_prime : Ideal.IsPrime.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) S], (Ne.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S (Bot.bot.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.hasBot.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))))) -> (Prime.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.IsPrincipal.generator.{u1, u1} R R (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2)))) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S _inst_4))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_2 : CommRing.{u1} R] (S : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) [_inst_4 : Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_2) (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S] [is_prime : Ideal.IsPrime.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) S], (Ne.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S (Bot.bot.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Submodule.instBotSubmodule.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)))))) -> (Prime.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2)) (Submodule.IsPrincipal.generator.{u1, u1} R R (Ring.toAddCommGroup.{u1} R (CommRing.toRing.{u1} R _inst_2)) (CommRing.toRing.{u1} R _inst_2) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2))) S _inst_4))
+Case conversion may be inaccurate. Consider using '#align submodule.is_principal.prime_generator_of_is_prime Submodule.IsPrincipal.prime_generator_of_isPrimeₓ'. -/
 theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_prime : S.IsPrime]
     (ne_bot : S ≠ ⊥) : Prime (generator S) :=
   ⟨fun h => ne_bot ((eq_bot_iff_generator_eq_zero S).2 h), fun h =>
@@ -136,6 +178,12 @@ theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_p
     simpa only [← mem_iff_generator_dvd S] using is_prime.2⟩
 #align submodule.is_principal.prime_generator_of_is_prime Submodule.IsPrincipal.prime_generator_of_isPrime
 
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
     (hx : x ∈ N) : generator (N.map ϕ) ∣ ϕ x :=
@@ -144,6 +192,12 @@ theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.ma
   exact ⟨x, hx, rfl⟩
 #align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_mem
 
+/- warning: submodule.is_principal.generator_submodule_image_dvd_of_mem -> Submodule.IsPrincipal.generator_submoduleImage_dvd_of_mem is a dubious translation:
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(AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) => R) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u2, u1} R R (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O)) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (Submodule.addCommMonoid.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Submodule.module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3 O) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_2) _inst_2 (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_2))) (RingHom.id.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) ϕ (Subtype.mk.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) (SetLike.instMembership.{u2, u2} (Submodule.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3) M (Submodule.setLike.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_2)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_1) _inst_3)) x O) x (hNO x hx)))
+Case conversion may be inaccurate. Consider using '#align submodule.is_principal.generator_submodule_image_dvd_of_mem Submodule.IsPrincipal.generator_submoduleImage_dvd_of_memₓ'. -/
 -- Note that the converse may not hold if `ϕ` is not injective.
 theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
     [(ϕ.submoduleImage N).IsPrincipal] {x : M} (hx : x ∈ N) :
@@ -161,6 +215,12 @@ namespace IsPrime
 
 open Submodule.IsPrincipal Ideal
 
+/- warning: is_prime.to_maximal_ideal -> IsPrime.to_maximal_ideal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {S : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} [hpi : Ideal.IsPrime.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) S], (Ne.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) S (Bot.bot.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasBot.{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.IsMaximal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) S)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {S : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} [hpi : Ideal.IsPrime.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) S], (Ne.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) S (Bot.bot.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.instBotSubmodule.{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.IsMaximal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) S)
+Case conversion may be inaccurate. Consider using '#align is_prime.to_maximal_ideal IsPrime.to_maximal_idealₓ'. -/
 -- TODO -- for a non-ID one could perhaps prove that if p < q are prime then q maximal;
 -- 0 isn't prime in a non-ID PIR but the Krull dimension is still <= 1.
 -- The below result follows from this, but we could also use the below result to
@@ -189,11 +249,14 @@ open EuclideanDomain
 
 variable [EuclideanDomain R]
 
+#print mod_mem_iff /-
 theorem mod_mem_iff {S : Ideal R} {x y : R} (hy : y ∈ S) : x % y ∈ S ↔ x ∈ S :=
   ⟨fun hxy => div_add_mod x y ▸ S.add_mem (S.mul_mem_right _ hy) hxy, fun hx =>
     (mod_eq_sub_mul_div x y).symm ▸ S.sub_mem hx (S.mul_mem_right _ hy)⟩
 #align mod_mem_iff mod_mem_iff
+-/
 
+#print EuclideanDomain.to_principal_ideal_domain /-
 -- see Note [lower instance priority]
 instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincipalIdealRing R
     where principal S :=
@@ -232,18 +295,22 @@ instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincip
                 ⟨fun haS => by_contradiction fun ha0 => h ⟨a, ⟨haS, ha0⟩⟩, fun h₁ =>
                   h₁.symm ▸ S.zero_mem⟩⟩⟩
 #align euclidean_domain.to_principal_ideal_domain EuclideanDomain.to_principal_ideal_domain
+-/
 
 end
 
+#print IsField.isPrincipalIdealRing /-
 theorem IsField.isPrincipalIdealRing {R : Type _} [CommRing R] (h : IsField R) :
     IsPrincipalIdealRing R :=
   @EuclideanDomain.to_principal_ideal_domain R (@Field.toEuclideanDomain R h.toField)
 #align is_field.is_principal_ideal_ring IsField.isPrincipalIdealRing
+-/
 
 namespace PrincipalIdealRing
 
 open IsPrincipalIdealRing
 
+#print PrincipalIdealRing.isNoetherianRing /-
 -- see Note [lower instance priority]
 instance (priority := 100) isNoetherianRing [Ring R] [IsPrincipalIdealRing R] :
     IsNoetherianRing R :=
@@ -254,7 +321,14 @@ instance (priority := 100) isNoetherianRing [Ring R] [IsPrincipalIdealRing R] :
       rw [← Finset.coe_singleton]
       exact ⟨{a}, SetLike.coe_injective rfl⟩⟩
 #align principal_ideal_ring.is_noetherian_ring PrincipalIdealRing.isNoetherianRing
+-/
 
+/- warning: principal_ideal_ring.is_maximal_of_irreducible -> PrincipalIdealRing.isMaximal_of_irreducible is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) -> (Ideal.IsMaximal.{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) p)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) -> (Ideal.IsMaximal.{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.instSingletonSet.{u1} R) p)))
+Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.is_maximal_of_irreducible PrincipalIdealRing.isMaximal_of_irreducibleₓ'. -/
 theorem isMaximal_of_irreducible [CommRing R] [IsPrincipalIdealRing R] {p : R}
     (hp : Irreducible p) : Ideal.IsMaximal (span R ({p} : Set R)) :=
   ⟨⟨mt Ideal.span_singleton_eq_top.1 hp.1, fun I hI =>
@@ -268,11 +342,23 @@ theorem isMaximal_of_irreducible [CommRing R] [IsPrincipalIdealRing R] {p : R}
 
 variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
 
+/- warning: principal_ideal_ring.irreducible_iff_prime -> PrincipalIdealRing.irreducible_iff_prime is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, Iff (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) (Prime.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) p)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : R}, Iff (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) (Prime.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)) p)
+Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.irreducible_iff_prime PrincipalIdealRing.irreducible_iff_primeₓ'. -/
 theorem irreducible_iff_prime {p : R} : Irreducible p ↔ Prime p :=
   ⟨fun hp => (Ideal.span_singleton_prime hp.NeZero).1 <| (isMaximal_of_irreducible hp).IsPrime,
     Prime.irreducible⟩
 #align principal_ideal_ring.irreducible_iff_prime PrincipalIdealRing.irreducible_iff_prime
 
+/- warning: principal_ideal_ring.associates_irreducible_iff_prime -> PrincipalIdealRing.associates_irreducible_iff_prime is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : Associates.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (Irreducible.{u1} (Associates.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) (MonoidWithZero.toMonoid.{u1} (Associates.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CommMonoidWithZero.toMonoidWithZero.{u1} (Associates.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Associates.commMonoidWithZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) p) (Prime.{u1} (Associates.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Associates.commMonoidWithZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) p)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] {p : Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))}, Iff (Irreducible.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (MonoidWithZero.toMonoid.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (CommMonoidWithZero.toMonoidWithZero.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))))) p) (Prime.{u1} (Associates.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Associates.instCommMonoidWithZeroAssociatesToMonoidToMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2))) p)
+Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.associates_irreducible_iff_prime PrincipalIdealRing.associates_irreducible_iff_primeₓ'. -/
 theorem associates_irreducible_iff_prime : ∀ {p : Associates R}, Irreducible p ↔ Prime p :=
   Associates.irreducible_iff_prime_iff.1 fun _ => irreducible_iff_prime
 #align principal_ideal_ring.associates_irreducible_iff_prime PrincipalIdealRing.associates_irreducible_iff_prime
@@ -281,11 +367,19 @@ section
 
 open Classical
 
+#print PrincipalIdealRing.factors /-
 /-- `factors a` is a multiset of irreducible elements whose product is `a`, up to units -/
 noncomputable def factors (a : R) : Multiset R :=
   if h : a = 0 then ∅ else Classical.choose (WfDvdMonoid.exists_factors a h)
 #align principal_ideal_ring.factors PrincipalIdealRing.factors
+-/
 
+/- warning: principal_ideal_ring.factors_spec -> PrincipalIdealRing.factors_spec is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] (a : R), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) -> (And (forall (b : R), (Membership.Mem.{u1, u1} R (Multiset.{u1} R) (Multiset.hasMem.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) -> (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) b)) (Associated.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Multiset.prod.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) a))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] (a : R), (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)))))) -> (And (forall (b : R), (Membership.mem.{u1, u1} R (Multiset.{u1} R) (Multiset.instMembershipMultiset.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) -> (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) b)) (Associated.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Multiset.prod.{u1} R (CommRing.toCommMonoid.{u1} R _inst_1) (PrincipalIdealRing.factors.{u1} R _inst_1 _inst_2 _inst_3 a)) a))
+Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.factors_spec PrincipalIdealRing.factors_specₓ'. -/
 theorem factors_spec (a : R) (h : a ≠ 0) :
     (∀ b ∈ factors a, Irreducible b) ∧ Associated (factors a).Prod a :=
   by
@@ -293,11 +387,23 @@ theorem factors_spec (a : R) (h : a ≠ 0) :
   exact Classical.choose_spec (WfDvdMonoid.exists_factors a h)
 #align principal_ideal_ring.factors_spec PrincipalIdealRing.factors_spec
 
+/- warning: principal_ideal_ring.ne_zero_of_mem_factors -> PrincipalIdealRing.ne_zero_of_mem_factors is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} [_inst_4 : CommRing.{u1} R] [_inst_5 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_4))] [_inst_6 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_4)] {a : R} {b : R}, (Ne.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_4) _inst_5)))))) -> (Membership.mem.{u1, u1} R (Multiset.{u1} R) (Multiset.instMembershipMultiset.{u1} R) b (PrincipalIdealRing.factors.{u1} R _inst_4 _inst_5 _inst_6 a)) -> (Ne.{succ u1} R b (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_4) _inst_5))))))
+Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.ne_zero_of_mem_factors PrincipalIdealRing.ne_zero_of_mem_factorsₓ'. -/
 theorem ne_zero_of_mem_factors {R : Type v} [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
     {a b : R} (ha : a ≠ 0) (hb : b ∈ factors a) : b ≠ 0 :=
   Irreducible.ne_zero ((factors_spec a ha).1 b hb)
 #align principal_ideal_ring.ne_zero_of_mem_factors PrincipalIdealRing.ne_zero_of_mem_factors
 
+/- warning: principal_ideal_ring.mem_submonoid_of_factors_subset_of_units_subset -> PrincipalIdealRing.mem_submonoid_of_factors_subset_of_units_subset is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.mem_submonoid_of_factors_subset_of_units_subsetₓ'. -/
 theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R} (ha : a ≠ 0)
     (hfac : ∀ b ∈ factors a, b ∈ s) (hunit : ∀ c : Rˣ, (c : R) ∈ s) : a ∈ s :=
   by
@@ -306,6 +412,12 @@ theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R
   exact mul_mem (multiset_prod_mem _ hfac) (hunit _)
 #align principal_ideal_ring.mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.mem_submonoid_of_factors_subset_of_units_subset
 
+/- warning: principal_ideal_ring.ring_hom_mem_submonoid_of_factors_subset_of_units_subset -> PrincipalIdealRing.ringHom_mem_submonoid_of_factors_subset_of_units_subset is a dubious translation:
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_inst_6 a)) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) b) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S 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(RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f b) s)) -> (forall (c : Units.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_4))))), Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_4)))) c)) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7) (RingHom.instRingHomClassRingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f (Units.val.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_4)))) c)) s) -> (Membership.mem.{u1, u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) a) (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) (SetLike.instMembership.{u1, u1} (Submonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7)))) S (Submonoid.instSetLikeSubmonoid.{u1} S (MulZeroOneClass.toMulOneClass.{u1} S (NonAssocSemiring.toMulZeroOneClass.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R S (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7)) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S 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R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_4))) (Semiring.toNonAssocSemiring.{u1} S _inst_7))))) f a) s)
+Case conversion may be inaccurate. Consider using '#align principal_ideal_ring.ring_hom_mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.ringHom_mem_submonoid_of_factors_subset_of_units_subsetₓ'. -/
 /-- If a `ring_hom` maps all units and all factors of an element `a` into a submonoid `s`, then it
 also maps `a` into that submonoid. -/
 theorem ringHom_mem_submonoid_of_factors_subset_of_units_subset {R S : Type _} [CommRing R]
@@ -314,12 +426,14 @@ theorem ringHom_mem_submonoid_of_factors_subset_of_units_subset {R S : Type _} [
   mem_submonoid_of_factors_subset_of_units_subset (s.comap f.toMonoidHom) ha h hf
 #align principal_ideal_ring.ring_hom_mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.ringHom_mem_submonoid_of_factors_subset_of_units_subset
 
+#print PrincipalIdealRing.to_uniqueFactorizationMonoid /-
 -- see Note [lower instance priority]
 /-- A principal ideal domain has unique factorization -/
 instance (priority := 100) to_uniqueFactorizationMonoid : UniqueFactorizationMonoid R :=
   { (IsNoetherianRing.wfDvdMonoid : WfDvdMonoid R) with
     irreducible_iff_prime := fun _ => PrincipalIdealRing.irreducible_iff_prime }
 #align principal_ideal_ring.to_unique_factorization_monoid PrincipalIdealRing.to_uniqueFactorizationMonoid
+-/
 
 end
 
@@ -333,6 +447,12 @@ variable {S N : Type _} [Ring R] [AddCommGroup M] [AddCommGroup N] [Ring S]
 
 variable [Module R M] [Module R N]
 
+/- warning: submodule.is_principal.of_comap -> Submodule.IsPrincipal.of_comap is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : AddCommGroup.{u3} N] [_inst_5 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] [_inst_6 : Module.{u1, u3} R N (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3)] (f : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u2, succ u3} M N (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) (fun (_x : LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) => M -> N) (LinearMap.hasCoeToFun.{u1, u1, u2, u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f)) -> (forall (S : Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u1, u2} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u1, u1, u2, u3, max u2 u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (LinearMap.{u1, u1, u2, u3} R R (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6) (LinearMap.semilinearMapClass.{u1, u1, u2, u3} R R M N (Ring.toSemiring.{u1} R _inst_1) (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) (AddCommGroup.toAddCommMonoid.{u3} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R _inst_1)))) f S)], Submodule.IsPrincipal.{u1, u3} R N _inst_1 _inst_3 _inst_6 S)
+but is expected to have type
+  forall {R : Type.{u2}} {M : Type.{u3}} {N : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : AddCommGroup.{u3} M] [_inst_3 : AddCommGroup.{u1} N] [_inst_5 : Module.{u2, u3} R M (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2)] [_inst_6 : Module.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3)] (f : LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6), (Function.Surjective.{succ u3, succ u1} M N (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => N) _x) (LinearMap.instFunLikeLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) f)) -> (forall (S : Submodule.{u2, u1} R N (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_6) [hI : Submodule.IsPrincipal.{u2, u3} R M _inst_1 _inst_2 _inst_5 (Submodule.comap.{u2, u2, u3, u1, max u3 u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) (LinearMap.{u2, u2, u3, u1} R R (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1))) M N (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6) (LinearMap.instSemilinearMapClassLinearMap.{u2, u2, u3, u1} R R M N (Ring.toSemiring.{u2} R _inst_1) (Ring.toSemiring.{u2} R _inst_1) (AddCommGroup.toAddCommMonoid.{u3} M _inst_2) (AddCommGroup.toAddCommMonoid.{u1} N _inst_3) _inst_5 _inst_6 (RingHom.id.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R _inst_1)))) f S)], Submodule.IsPrincipal.{u2, u1} R N _inst_1 _inst_3 _inst_6 S)
+Case conversion may be inaccurate. Consider using '#align submodule.is_principal.of_comap Submodule.IsPrincipal.of_comapₓ'. -/
 theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjective f)
     (S : Submodule R N) [hI : IsPrincipal (S.comap f)] : IsPrincipal S :=
   ⟨⟨f (IsPrincipal.generator (S.comap f)), by
@@ -340,6 +460,12 @@ theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjec
         Submodule.map_comap_eq_of_surjective hf]⟩⟩
 #align submodule.is_principal.of_comap Submodule.IsPrincipal.of_comap
 
+/- warning: ideal.is_principal.of_comap -> Ideal.IsPrincipal.of_comap 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.is_principal.of_comap Ideal.IsPrincipal.of_comapₓ'. -/
 theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f) (I : Ideal S)
     [hI : IsPrincipal (I.comap f)] : IsPrincipal I :=
   ⟨⟨f (IsPrincipal.generator (I.comap f)), by
@@ -347,6 +473,12 @@ theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f)
         Ideal.span_singleton_generator, Ideal.map_comap_of_surjective f hf]⟩⟩
 #align ideal.is_principal.of_comap Ideal.IsPrincipal.of_comap
 
+/- warning: is_principal_ideal_ring.of_surjective -> IsPrincipalIdealRing.of_surjective is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_4 : Ring.{u2} S] [_inst_7 : IsPrincipalIdealRing.{u1} R _inst_1] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))), (Function.Surjective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R _inst_1)) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S _inst_4))) f)) -> (IsPrincipalIdealRing.{u2} S _inst_4)
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align is_principal_ideal_ring.of_surjective IsPrincipalIdealRing.of_surjectiveₓ'. -/
 /-- The surjective image of a principal ideal ring is again a principal ideal ring. -/
 theorem IsPrincipalIdealRing.of_surjective [IsPrincipalIdealRing R] (f : R →+* S)
     (hf : Function.Surjective f) : IsPrincipalIdealRing S :=
@@ -361,6 +493,7 @@ open Ideal
 
 variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [GCDMonoid R]
 
+#print span_gcd /-
 theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
   by
   obtain ⟨d, hd⟩ := IsPrincipalIdealRing.principal (span ({x, y} : Set R))
@@ -382,21 +515,46 @@ theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
     apply dvd_add <;> apply dvd_mul_of_dvd_right
     exacts[gcd_dvd_left x y, gcd_dvd_right x y]
 #align span_gcd span_gcd
+-/
 
+/- warning: gcd_dvd_iff_exists -> gcd_dvd_iff_exists is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align gcd_dvd_iff_exists gcd_dvd_iff_existsₓ'. -/
 theorem gcd_dvd_iff_exists (a b : R) {z} : gcd a b ∣ z ↔ ∃ x y, z = a * x + b * y := by
   simp_rw [mul_comm a, mul_comm b, @eq_comm _ z, ← Ideal.mem_span_pair, ← span_gcd,
     Ideal.mem_span_singleton]
 #align gcd_dvd_iff_exists gcd_dvd_iff_exists
 
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+Case conversion may be inaccurate. Consider using '#align exists_gcd_eq_mul_add_mul exists_gcd_eq_mul_add_mulₓ'. -/
 /-- **Bézout's lemma** -/
 theorem exists_gcd_eq_mul_add_mul (a b : R) : ∃ x y, gcd a b = a * x + b * y := by
   rw [← gcd_dvd_iff_exists]
 #align exists_gcd_eq_mul_add_mul exists_gcd_eq_mul_add_mul
 
+/- warning: gcd_is_unit_iff -> gcd_isUnit_iff is a dubious translation:
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+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), Iff (IsUnit.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 x y)) (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), Iff (IsUnit.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (GCDMonoid.gcd.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2) _inst_4 x y)) (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y)
+Case conversion may be inaccurate. Consider using '#align gcd_is_unit_iff gcd_isUnit_iffₓ'. -/
 theorem gcd_isUnit_iff (x y : R) : IsUnit (gcd x y) ↔ IsCoprime x y := by
   rw [IsCoprime, ← Ideal.mem_span_pair, ← span_gcd, ← span_singleton_eq_top, eq_top_iff_one]
 #align gcd_is_unit_iff gcd_isUnit_iff
 
+/- warning: is_coprime_of_dvd -> isCoprime_of_dvd is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_coprime_of_dvd isCoprime_of_dvdₓ'. -/
 -- this should be proved for UFDs surely?
 theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z ∈ nonunits R, z ≠ 0 → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
@@ -407,6 +565,12 @@ theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
   rwa [Ne, gcd_eq_zero_iff]
 #align is_coprime_of_dvd isCoprime_of_dvd
 
+/- warning: dvd_or_coprime -> dvd_or_coprime is a dubious translation:
+lean 3 declaration is
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+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] (x : R) (y : R), (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) x) -> (Or (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) x y) (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) x y))
+Case conversion may be inaccurate. Consider using '#align dvd_or_coprime dvd_or_coprimeₓ'. -/
 -- this should be proved for UFDs surely?
 theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y :=
   by
@@ -419,6 +583,12 @@ theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y
     simpa using mul_dvd_mul_left z (isUnit_iff_dvd_one.1 <| (of_irreducible_mul h).resolve_left nu)
 #align dvd_or_coprime dvd_or_coprime
 
+/- warning: is_coprime_of_irreducible_dvd -> isCoprime_of_irreducible_dvd is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align is_coprime_of_irreducible_dvd isCoprime_of_irreducible_dvdₓ'. -/
 theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z : R, Irreducible z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
   by
@@ -430,11 +600,23 @@ theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
       assumption
 #align is_coprime_of_irreducible_dvd isCoprime_of_irreducible_dvd
 
+/- warning: is_coprime_of_prime_dvd -> isCoprime_of_prime_dvd is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_coprime_of_prime_dvd isCoprime_of_prime_dvdₓ'. -/
 theorem isCoprime_of_prime_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z : R, Prime z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
   isCoprime_of_irreducible_dvd nonzero fun z zi => H z <| GCDMonoid.prime_of_irreducible zi
 #align is_coprime_of_prime_dvd isCoprime_of_prime_dvd
 
+/- warning: irreducible.coprime_iff_not_dvd -> Irreducible.coprime_iff_not_dvd is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {n : R}, (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) -> (Iff (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p n) (Not (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p n)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {n : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) -> (Iff (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p n) (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p n)))
+Case conversion may be inaccurate. Consider using '#align irreducible.coprime_iff_not_dvd Irreducible.coprime_iff_not_dvdₓ'. -/
 theorem Irreducible.coprime_iff_not_dvd {p n : R} (pp : Irreducible p) : IsCoprime p n ↔ ¬p ∣ n :=
   by
   constructor
@@ -452,23 +634,49 @@ theorem Irreducible.coprime_iff_not_dvd {p n : R} (pp : Irreducible p) : IsCopri
     exact nd ((zi.associated_of_dvd pp zp).symm.Dvd.trans zn)
 #align irreducible.coprime_iff_not_dvd Irreducible.coprime_iff_not_dvd
 
+#print Prime.coprime_iff_not_dvd /-
 theorem Prime.coprime_iff_not_dvd {p n : R} (pp : Prime p) : IsCoprime p n ↔ ¬p ∣ n :=
   pp.Irreducible.coprime_iff_not_dvd
 #align prime.coprime_iff_not_dvd Prime.coprime_iff_not_dvd
+-/
 
+/- warning: irreducible.dvd_iff_not_coprime -> Irreducible.dvd_iff_not_coprime is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {n : R}, (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) -> (Iff (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p n) (Not (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p n)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {n : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) -> (Iff (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p n) (Not (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p n)))
+Case conversion may be inaccurate. Consider using '#align irreducible.dvd_iff_not_coprime Irreducible.dvd_iff_not_coprimeₓ'. -/
 theorem Irreducible.dvd_iff_not_coprime {p n : R} (hp : Irreducible p) : p ∣ n ↔ ¬IsCoprime p n :=
   iff_not_comm.2 hp.coprime_iff_not_dvd
 #align irreducible.dvd_iff_not_coprime Irreducible.dvd_iff_not_coprime
 
+/- warning: irreducible.coprime_pow_of_not_dvd -> Irreducible.coprime_pow_of_not_dvd is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {a : R} (m : Nat), (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) -> (Not (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p a)) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) a (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p m))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R} {a : R} (m : Nat), (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) -> (Not (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p a)) -> (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) a (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 (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) p m))
+Case conversion may be inaccurate. Consider using '#align irreducible.coprime_pow_of_not_dvd Irreducible.coprime_pow_of_not_dvdₓ'. -/
 theorem Irreducible.coprime_pow_of_not_dvd {p a : R} (m : ℕ) (hp : Irreducible p) (h : ¬p ∣ a) :
     IsCoprime a (p ^ m) :=
   (hp.coprime_iff_not_dvd.2 h).symm.pow_right
 #align irreducible.coprime_pow_of_not_dvd Irreducible.coprime_pow_of_not_dvd
 
+/- warning: irreducible.coprime_or_dvd -> Irreducible.coprime_or_dvd is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R}, (Irreducible.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) -> (forall (i : R), Or (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p i) (Dvd.Dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p i))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {p : R}, (Irreducible.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) p) -> (forall (i : R), Or (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) p i) (Dvd.dvd.{u1} R (semigroupDvd.{u1} R (SemigroupWithZero.toSemigroup.{u1} R (NonUnitalSemiring.toSemigroupWithZero.{u1} R (NonUnitalRing.toNonUnitalSemiring.{u1} R (NonUnitalCommRing.toNonUnitalRing.{u1} R (CommRing.toNonUnitalCommRing.{u1} R _inst_1)))))) p i))
+Case conversion may be inaccurate. Consider using '#align irreducible.coprime_or_dvd Irreducible.coprime_or_dvdₓ'. -/
 theorem Irreducible.coprime_or_dvd {p : R} (hp : Irreducible p) (i : R) : IsCoprime p i ∨ p ∣ i :=
   (em _).imp_right hp.dvd_iff_not_coprime.2
 #align irreducible.coprime_or_dvd Irreducible.coprime_or_dvd
 
+/- warning: exists_associated_pow_of_mul_eq_pow' -> exists_associated_pow_of_mul_eq_pow' is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {a : R} {b : R} {c : R}, (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) a b) -> (forall {k : Nat}, (Eq.{succ 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) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c k)) -> (Exists.{succ u1} R (fun (d : R) => Associated.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) d k) a)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] [_inst_2 : IsDomain.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] [_inst_3 : IsPrincipalIdealRing.{u1} R (CommRing.toRing.{u1} R _inst_1)] [_inst_4 : GCDMonoid.{u1} R (IsDomain.toCancelCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2)] {a : R} {b : R} {c : R}, (IsCoprime.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1) a b) -> (forall {k : Nat}, (Eq.{succ 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) (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 (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) c k)) -> (Exists.{succ u1} R (fun (d : R) => Associated.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (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 (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) d k) a)))
+Case conversion may be inaccurate. Consider using '#align exists_associated_pow_of_mul_eq_pow' exists_associated_pow_of_mul_eq_pow'ₓ'. -/
 theorem exists_associated_pow_of_mul_eq_pow' {a b c : R} (hab : IsCoprime a b) {k : ℕ}
     (h : a * b = c ^ k) : ∃ d, Associated (d ^ k) a :=
   exists_associated_pow_of_mul_eq_pow ((gcd_isUnit_iff _ _).mpr hab) h
@@ -482,21 +690,33 @@ open Set Ideal
 
 variable (R) [CommRing R]
 
+#print nonPrincipals /-
 /-- `non_principals R` is the set of all ideals of `R` that are not principal ideals. -/
 def nonPrincipals :=
   { I : Ideal R | ¬I.IsPrincipal }
 #align non_principals nonPrincipals
+-/
 
+#print nonPrincipals_def /-
 theorem nonPrincipals_def {I : Ideal R} : I ∈ nonPrincipals R ↔ ¬I.IsPrincipal :=
   Iff.rfl
 #align non_principals_def nonPrincipals_def
+-/
 
 variable {R}
 
+#print nonPrincipals_eq_empty_iff /-
 theorem nonPrincipals_eq_empty_iff : nonPrincipals R = ∅ ↔ IsPrincipalIdealRing R := by
   simp [Set.eq_empty_iff_forall_not_mem, isPrincipalIdealRing_iff, nonPrincipals_def]
 #align non_principals_eq_empty_iff nonPrincipals_eq_empty_iff
+-/
 
+/- warning: non_principals_zorn -> nonPrincipals_zorn is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasSubset.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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)))))))) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => Exists.{0} (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (fun (H : Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) => forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.Mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.hasMem.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{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))))))) J I)))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (c : Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instHasSubsetSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) c (nonPrincipals.{u1} R _inst_1)) -> (IsChain.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4580 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4582 : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4580 x._@.Mathlib.RingTheory.PrincipalIdealDomain._hyg.4582) c) -> (forall {K : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) K c) -> (Exists.{succ u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (fun (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) => And (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) I (nonPrincipals.{u1} R _inst_1)) (forall (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), (Membership.mem.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Set.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Set.instMembershipSet.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))) J c) -> (LE.le.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Preorder.toLE.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.completeLattice.{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)))))))) J I)))))
+Case conversion may be inaccurate. Consider using '#align non_principals_zorn nonPrincipals_zornₓ'. -/
 /-- Any chain in the set of non-principal ideals has an upper bound which is non-principal.
 (Namely, the union of the chain is such an upper bound.)
 -/
@@ -514,6 +734,7 @@ theorem nonPrincipals_zorn (c : Set (Ideal R)) (hs : c ⊆ nonPrincipals R)
   exact hs ⟨⟨x, rfl⟩⟩
 #align non_principals_zorn nonPrincipals_zorn
 
+#print IsPrincipalIdealRing.of_prime /-
 /-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
 theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
     IsPrincipalIdealRing R :=
@@ -564,6 +785,7 @@ theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPr
       span_singleton_mul_span_singleton, mul_comm y, Ideal.span_singleton_le_iff_mem]
     exact ⟨mul_le_right, Ideal.mem_colon_singleton.1 <| hb.symm ▸ Ideal.mem_span_singleton_self b⟩
 #align is_principal_ideal_ring.of_prime IsPrincipalIdealRing.of_prime
+-/
 
 end PrincipalOfPrime
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/3180fab693e2cee3bff62675571264cb8778b212

@@ -374,9 +374,9 @@ theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
   · rw [dvd_gcd_iff]
     constructor <;> rw [← Ideal.mem_span_singleton, ← hd, Ideal.mem_span_pair]
     · use 1, 0
-      rw [one_mul, zero_mul, add_zero]
+      rw [one_mul, MulZeroClass.zero_mul, add_zero]
     · use 0, 1
-      rw [one_mul, zero_mul, zero_add]
+      rw [one_mul, MulZeroClass.zero_mul, zero_add]
   · obtain ⟨r, s, rfl⟩ : ∃ r s, r * x + s * y = d := by
       rw [← Ideal.mem_span_pair, hd, Ideal.mem_span_singleton]
     apply dvd_add <;> apply dvd_mul_of_dvd_right

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

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

@@ -71,8 +71,8 @@ instance (priority := 100) DivisionRing.isPrincipalIdealRing (K : Type u) [Divis
     IsPrincipalIdealRing K where
   principal S := by
     rcases Ideal.eq_bot_or_top S with (rfl | rfl)
-    apply bot_isPrincipal
-    apply top_isPrincipal
+    · apply bot_isPrincipal
+    · apply top_isPrincipal
 #align division_ring.is_principal_ideal_ring DivisionRing.isPrincipalIdealRing
 
 end

This is based on seeing the import RingTheory.Ideal.Operations → LinearAlgebra.Basis on the longest pole. It feels like Ideal.Operations is a bit of a chokepoint for compiling Mathlib since it imports many files and is imported by many files. So splitting out a few obvious parts should help with compile times. Moreover, there are a bunch of imports that I could remove and have the file still compile: presumably these are (were) transitive dependencies that shake does not remove.

The following results and their corollaries were split off:

• Ideal.basisSpanSingleton
• Basis.mem_ideal_iff
• Ideal.colon

In particular, now Ideal.Operations should no longer need to know about Basis or submodule quotients.

@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Morenikeji Neri
 -/
 import Mathlib.Algebra.EuclideanDomain.Instances
+import Mathlib.RingTheory.Ideal.Colon
 import Mathlib.RingTheory.UniqueFactorizationDomain
 
 #align_import ring_theory.principal_ideal_domain from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"

@@ -161,8 +161,8 @@ section
 
 variable [Ring R]
 
-instance span_pair_isPrincipal [IsBezout R] (x y : R) : (Ideal.span {x, y}).IsPrincipal :=
-  by classical exact isPrincipal_of_FG (Ideal.span {x, y}) ⟨{x, y}, by simp⟩
+instance span_pair_isPrincipal [IsBezout R] (x y : R) : (Ideal.span {x, y}).IsPrincipal := by
+  classical exact isPrincipal_of_FG (Ideal.span {x, y}) ⟨{x, y}, by simp⟩
 #align is_bezout.span_pair_is_principal IsBezout.span_pair_isPrincipal
 
 variable (x y : R) [(Ideal.span {x, y}).IsPrincipal]
@@ -212,7 +212,7 @@ theorem _root_.isRelPrime_iff_isCoprime : IsRelPrime x y ↔ IsCoprime x y :=
 
 variable (R)
 
-/-- Any bezout domain is a GCD domain. This is not an instance since `GCDMonoid` contains data,
+/-- Any Bézout domain is a GCD domain. This is not an instance since `GCDMonoid` contains data,
 and this might not be how we would like to construct it. -/
 noncomputable def toGCDDomain [IsBezout R] [IsDomain R] [DecidableEq R] : GCDMonoid R :=
   gcdMonoidOfGCD (gcd · ·) (gcd_dvd_left · ·) (gcd_dvd_right · ·) dvd_gcd

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)

@@ -379,7 +379,6 @@ section Surjective
 open Submodule
 
 variable {S N : Type*} [Ring R] [AddCommGroup M] [AddCommGroup N] [Ring S]
-
 variable [Module R M] [Module R N]
 
 theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjective f)

We remove all but one open Classicals, instead preferring to use open scoped Classical. The only real side-effect this led to is moving a couple declarations to use Exists.choose instead of Classical.choose.

The first few commits are explicitly labelled regex replaces for ease of review.

@@ -330,7 +330,7 @@ variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
 
 section
 
-open Classical
+open scoped Classical
 
 /-- `factors a` is a multiset of irreducible elements whose product is `a`, up to units -/
 noncomputable def factors (a : R) : Multiset R :=

• 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>

@@ -29,7 +29,7 @@ Theorems about PID's are in the `principal_ideal_ring` namespace.
 
 - `to_maximal_ideal`: a non-zero prime ideal in a PID is maximal.
 - `EuclideanDomain.to_principal_ideal_domain` : a Euclidean domain is a PID.
-- `IsBezout.toGCDDomain`: Every Bézout domain is a GCD domain. This is not an instance.
+- `IsBezout.nonemptyGCDMonoid`: Every Bézout domain is a GCD domain.
 
 -/
 
@@ -46,18 +46,6 @@ section
 
 variable [Ring R] [AddCommGroup M] [Module R M]
 
--- Porting note: renamed field to `principal'` and added `principal` to fix explicit argument
-/-- An `R`-submodule of `M` is principal if it is generated by one element. -/
-@[mk_iff]
-class Submodule.IsPrincipal (S : Submodule R M) : Prop where
-  principal' : ∃ a, S = span R {a}
-#align submodule.is_principal Submodule.IsPrincipal
-
-theorem Submodule.IsPrincipal.principal (S : Submodule R M) [S.IsPrincipal] :
-    ∃ a, S = span R {a} :=
-  Submodule.IsPrincipal.principal'
-#align submodule.is_principal.principal Submodule.IsPrincipal.principal
-
 instance bot_isPrincipal : (⊥ : Submodule R M).IsPrincipal :=
   ⟨⟨0, by simp⟩⟩
 #align bot_is_principal bot_isPrincipal
@@ -68,14 +56,6 @@ instance top_isPrincipal : (⊤ : Submodule R R).IsPrincipal :=
 
 variable (R)
 
-/-- A ring is a principal ideal ring if all (left) ideals are principal. -/
-@[mk_iff]
-class IsPrincipalIdealRing (R : Type u) [Ring R] : Prop where
-  principal : ∀ S : Ideal R, S.IsPrincipal
-#align is_principal_ideal_ring IsPrincipalIdealRing
-
-attribute [instance] IsPrincipalIdealRing.principal
-
 /-- A Bézout ring is a ring whose finitely generated ideals are principal. -/
 class IsBezout : Prop where
   /-- Any finitely generated ideal is principal. -/
@@ -238,7 +218,8 @@ noncomputable def toGCDDomain [IsBezout R] [IsDomain R] [DecidableEq R] : GCDMon
   gcdMonoidOfGCD (gcd · ·) (gcd_dvd_left · ·) (gcd_dvd_right · ·) dvd_gcd
 #align is_bezout.to_gcd_domain IsBezout.toGCDDomain
 
-instance [IsBezout R] [IsDomain R] : Nonempty (GCDMonoid R) := by classical exact ⟨toGCDDomain R⟩
+instance nonemptyGCDMonoid [IsBezout R] [IsDomain R] : Nonempty (GCDMonoid R) := by
+  classical exact ⟨toGCDDomain R⟩
 
 end IsBezout
 

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>

@@ -258,8 +258,8 @@ theorem to_maximal_ideal [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] {S :
       cases' (mem_iff_generator_dvd _).1 (hST <| generator_mem S) with z hz
       cases hpi.mem_or_mem (show generator T * z ∈ S from hz ▸ generator_mem S) with
       | inl h =>
-        have hTS : T ≤ S
-        rwa [← T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]
+        have hTS : T ≤ S := by
+          rwa [← T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]
         exact (hxS <| hTS hxT).elim
       | inr h =>
         cases' (mem_iff_generator_dvd _).1 h with y hy
@@ -574,8 +574,8 @@ theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPr
   obtain ⟨a, ha⟩ : (I ⊔ span {y}).IsPrincipal :=
     Imax' (left_lt_sup.mpr (mt I.span_singleton_le_iff_mem.mp hy))
   -- Then `x ∈ I.colon (span {y})`, which is equal to `I` if it's not principal.
-  suffices He : ¬(I.colon (span {y})).IsPrincipal
-  · rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
+  suffices He : ¬(I.colon (span {y})).IsPrincipal by
+    rw [← Imax _ ((nonPrincipals_def R).2 He) fun a ha =>
         Ideal.mem_colon_singleton.2 (mul_mem_right _ _ ha)]
     exact Ideal.mem_colon_singleton.2 hxy
   -- So suppose for the sake of contradiction that both `I ⊔ span {y}` and `I.colon (span {y})`

• Introduce typeclass DecompositionMonoid, which says every element in the monoid is primal, i.e., whenever an element divides a product b * c, it can be factored into a product such that the factors divides b and c respectively. A domain is called pre-Schreier if its multiplicative monoid is a decomposition monoid, and these are more general than GCD domains.

• Show that any GCDMonoid is a DecompositionMonoid. In order for lemmas about DecompositionMonoids to automatically apply to UniqueFactorizationMonoids, we add instances from UniqueFactorizationMonoid α to Nonempty (NormalizedGCDMonoid α) to Nonempty (GCDMonoid α) to DecompositionMonoid α. (Zulip) See the bottom of message for an updated diagram of classes and instances.

• Introduce binary predicate IsRelPrime which says that the only common divisors of the two elements are units. Replace previous occurrences in mathlib by this predicate.

• Duplicate all lemmas about IsCoprime in Coprime/Basic (except three lemmas about smul) to IsRelPrime. Due to import constraints, they are spread into three files Algebra/Divisibility/Units (including key lemmas assuming DecompositionMonoid), GroupWithZero/Divisibility, and Coprime/Basic.

• Show IsCoprime always imply IsRelPrime and is equivalent to it in Bezout rings. To reduce duplication, the definition of Bezout rings and the GCDMonoid instance are moved from RingTheory/Bezout to RingTheory/PrincipalIdealDomain, and some results in PrincipalIdealDomain are generalized to Bezout rings.

• Remove the recently added file Squarefree/UniqueFactorizationMonoid and place the results appropriately within Squarefree/Basic. All results are generalized to DecompositionMonoid or weaker except the last one.

Zulip

With this PR, all the following instances (indicated by arrows) now work; this PR fills the central part.

                                                                          EuclideanDomain (bundled)
                                                                              ↙          ↖
                                                                 IsPrincipalIdealRing ← Field (bundled)
                                                                            ↓             ↓
         NormalizationMonoid ←          NormalizedGCDMonoid → GCDMonoid  IsBezout ← ValuationRing ← DiscreteValuationRing
                   ↓                             ↓                 ↘       ↙
Nonempty NormalizationMonoid ← Nonempty NormalizedGCDMonoid →  Nonempty GCDMonoid → IsIntegrallyClosed
                                                 ↑                    ↓
                    WfDvdMonoid ← UniqueFactorizationMonoid → DecompositionMonoid
                                                 ↑
                                       IsPrincipalIdealRing

Co-authored-by: Junyan Xu <junyanxu.math@gmail.com> Co-authored-by: Oliver Nash <github@olivernash.org>

@@ -9,7 +9,7 @@ import Mathlib.RingTheory.UniqueFactorizationDomain
 #align_import ring_theory.principal_ideal_domain from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
 
 /-!
-# Principal ideal rings and principal ideal domains
+# Principal ideal rings, principal ideal domains, and Bézout rings
 
 A principal ideal ring (PIR) is a ring in which all left ideals are principal. A
 principal ideal domain (PID) is an integral domain which is a principal ideal ring.
@@ -21,6 +21,7 @@ Note that for principal ideal domains, one should use
 Theorems about PID's are in the `principal_ideal_ring` namespace.
 
 - `IsPrincipalIdealRing`: a predicate on rings, saying that every left ideal is principal.
+- `IsBezout`: the predicate saying that every finitely generated left ideal is principal.
 - `generator`: a generator of a principal ideal (or more generally submodule)
 - `to_unique_factorization_monoid`: a PID is a unique factorization domain
 
@@ -28,6 +29,7 @@ Theorems about PID's are in the `principal_ideal_ring` namespace.
 
 - `to_maximal_ideal`: a non-zero prime ideal in a PID is maximal.
 - `EuclideanDomain.to_principal_ideal_domain` : a Euclidean domain is a PID.
+- `IsBezout.toGCDDomain`: Every Bézout domain is a GCD domain. This is not an instance.
 
 -/
 
@@ -40,8 +42,6 @@ open Set Function
 
 open Submodule
 
-open Classical
-
 section
 
 variable [Ring R] [AddCommGroup M] [Module R M]
@@ -53,7 +53,7 @@ class Submodule.IsPrincipal (S : Submodule R M) : Prop where
   principal' : ∃ a, S = span R {a}
 #align submodule.is_principal Submodule.IsPrincipal
 
-theorem Submodule.IsPrincipal.principal (S : Submodule R M) [Submodule.IsPrincipal S] :
+theorem Submodule.IsPrincipal.principal (S : Submodule R M) [S.IsPrincipal] :
     ∃ a, S = span R {a} :=
   Submodule.IsPrincipal.principal'
 #align submodule.is_principal.principal Submodule.IsPrincipal.principal
@@ -76,6 +76,16 @@ class IsPrincipalIdealRing (R : Type u) [Ring R] : Prop where
 
 attribute [instance] IsPrincipalIdealRing.principal
 
+/-- A Bézout ring is a ring whose finitely generated ideals are principal. -/
+class IsBezout : Prop where
+  /-- Any finitely generated ideal is principal. -/
+  isPrincipal_of_FG : ∀ I : Ideal R, I.FG → I.IsPrincipal
+#align is_bezout IsBezout
+
+instance (priority := 100) IsBezout.of_isPrincipalIdealRing [IsPrincipalIdealRing R] : IsBezout R :=
+  ⟨fun I _ => IsPrincipalIdealRing.principal I⟩
+#align is_bezout.of_is_principal_ideal_ring IsBezout.of_isPrincipalIdealRing
+
 instance (priority := 100) DivisionRing.isPrincipalIdealRing (K : Type u) [DivisionRing K] :
     IsPrincipalIdealRing K where
   principal S := by
@@ -139,7 +149,7 @@ theorem mem_iff_generator_dvd (S : Ideal R) [S.IsPrincipal] {x : R} : x ∈ S 
   (mem_iff_eq_smul_generator S).trans (exists_congr fun a => by simp only [mul_comm, smul_eq_mul])
 #align submodule.is_principal.mem_iff_generator_dvd Submodule.IsPrincipal.mem_iff_generator_dvd
 
-theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_prime : S.IsPrime]
+theorem prime_generator_of_isPrime (S : Ideal R) [S.IsPrincipal] [is_prime : S.IsPrime]
     (ne_bot : S ≠ ⊥) : Prime (generator S) :=
   ⟨fun h => ne_bot ((eq_bot_iff_generator_eq_zero S).2 h), fun h =>
     is_prime.ne_top (S.eq_top_of_isUnit_mem (generator_mem S) h), fun _ _ => by
@@ -165,6 +175,73 @@ end CommRing
 
 end Submodule.IsPrincipal
 
+namespace IsBezout
+
+section
+
+variable [Ring R]
+
+instance span_pair_isPrincipal [IsBezout R] (x y : R) : (Ideal.span {x, y}).IsPrincipal :=
+  by classical exact isPrincipal_of_FG (Ideal.span {x, y}) ⟨{x, y}, by simp⟩
+#align is_bezout.span_pair_is_principal IsBezout.span_pair_isPrincipal
+
+variable (x y : R) [(Ideal.span {x, y}).IsPrincipal]
+
+/-- A choice of gcd of two elements in a Bézout domain.
+
+Note that the choice is usually not unique. -/
+noncomputable def gcd : R := Submodule.IsPrincipal.generator (Ideal.span {x, y})
+#align is_bezout.gcd IsBezout.gcd
+
+theorem span_gcd : Ideal.span {gcd x y} = Ideal.span {x, y} :=
+  Ideal.span_singleton_generator _
+#align is_bezout.span_gcd IsBezout.span_gcd
+
+end
+
+variable [CommRing R] (x y z : R) [(Ideal.span {x, y}).IsPrincipal]
+
+theorem gcd_dvd_left : gcd x y ∣ x :=
+  (Submodule.IsPrincipal.mem_iff_generator_dvd _).mp (Ideal.subset_span (by simp))
+#align is_bezout.gcd_dvd_left IsBezout.gcd_dvd_left
+
+theorem gcd_dvd_right : gcd x y ∣ y :=
+  (Submodule.IsPrincipal.mem_iff_generator_dvd _).mp (Ideal.subset_span (by simp))
+#align is_bezout.gcd_dvd_right IsBezout.gcd_dvd_right
+
+variable {x y z} in
+theorem dvd_gcd (hx : z ∣ x) (hy : z ∣ y) : z ∣ gcd x y := by
+  rw [← Ideal.span_singleton_le_span_singleton] at hx hy ⊢
+  rw [span_gcd, Ideal.span_insert, sup_le_iff]
+  exact ⟨hx, hy⟩
+#align is_bezout.dvd_gcd IsBezout.dvd_gcd
+
+theorem gcd_eq_sum : ∃ a b : R, a * x + b * y = gcd x y :=
+  Ideal.mem_span_pair.mp (by rw [← span_gcd]; apply Ideal.subset_span; simp)
+#align is_bezout.gcd_eq_sum IsBezout.gcd_eq_sum
+
+variable {x y}
+
+theorem _root_.IsRelPrime.isCoprime (h : IsRelPrime x y) : IsCoprime x y := by
+  rw [← Ideal.isCoprime_span_singleton_iff, Ideal.isCoprime_iff_sup_eq, ← Ideal.span_union,
+    Set.singleton_union, ← span_gcd, Ideal.span_singleton_eq_top]
+  exact h (gcd_dvd_left x y) (gcd_dvd_right x y)
+
+theorem _root_.isRelPrime_iff_isCoprime : IsRelPrime x y ↔ IsCoprime x y :=
+  ⟨IsRelPrime.isCoprime, IsCoprime.isRelPrime⟩
+
+variable (R)
+
+/-- Any bezout domain is a GCD domain. This is not an instance since `GCDMonoid` contains data,
+and this might not be how we would like to construct it. -/
+noncomputable def toGCDDomain [IsBezout R] [IsDomain R] [DecidableEq R] : GCDMonoid R :=
+  gcdMonoidOfGCD (gcd · ·) (gcd_dvd_left · ·) (gcd_dvd_right · ·) dvd_gcd
+#align is_bezout.to_gcd_domain IsBezout.toGCDDomain
+
+instance [IsBezout R] [IsDomain R] : Nonempty (GCDMonoid R) := by classical exact ⟨toGCDDomain R⟩
+
+end IsBezout
+
 namespace IsPrime
 
 open Submodule.IsPrincipal Ideal
@@ -206,7 +283,7 @@ theorem mod_mem_iff {S : Ideal R} {x y : R} (hy : y ∈ S) : x % y ∈ S ↔ x 
 
 -- see Note [lower instance priority]
 instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincipalIdealRing R where
-  principal S :=
+  principal S := by classical exact
     ⟨if h : { x : R | x ∈ S ∧ x ≠ 0 }.Nonempty then
         have wf : WellFounded (EuclideanDomain.r : R → R → Prop) := EuclideanDomain.r_wellFounded
         have hmin : WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h ∈ S ∧
@@ -263,16 +340,12 @@ theorem isMaximal_of_irreducible [CommRing R] [IsPrincipalIdealRing R] {p : R}
       erw [Ideal.span_singleton_le_span_singleton, IsUnit.mul_right_dvd hb]⟩⟩
 #align principal_ideal_ring.is_maximal_of_irreducible PrincipalIdealRing.isMaximal_of_irreducible
 
-variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
+@[deprecated] protected alias irreducible_iff_prime := irreducible_iff_prime
+#align principal_ideal_ring.irreducible_iff_prime irreducible_iff_prime
+@[deprecated] protected alias associates_irreducible_iff_prime := associates_irreducible_iff_prime
+#align principal_ideal_ring.associates_irreducible_iff_prime associates_irreducible_iff_prime
 
-theorem irreducible_iff_prime {p : R} : Irreducible p ↔ Prime p :=
-  ⟨fun hp => (Ideal.span_singleton_prime hp.ne_zero).1 <| (isMaximal_of_irreducible hp).isPrime,
-    Prime.irreducible⟩
-#align principal_ideal_ring.irreducible_iff_prime PrincipalIdealRing.irreducible_iff_prime
-
-theorem associates_irreducible_iff_prime : ∀ {p : Associates R}, Irreducible p ↔ Prime p :=
-  fun {p} => (Associates.irreducible_iff_prime_iff.1 fun _ => irreducible_iff_prime) p
-#align principal_ideal_ring.associates_irreducible_iff_prime PrincipalIdealRing.associates_irreducible_iff_prime
+variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
 
 section
 
@@ -313,7 +386,7 @@ theorem ringHom_mem_submonoid_of_factors_subset_of_units_subset {R S : Type*} [C
 /-- A principal ideal domain has unique factorization -/
 instance (priority := 100) to_uniqueFactorizationMonoid : UniqueFactorizationMonoid R :=
   { (IsNoetherianRing.wfDvdMonoid : WfDvdMonoid R) with
-    irreducible_iff_prime := PrincipalIdealRing.irreducible_iff_prime }
+    irreducible_iff_prime := irreducible_iff_prime }
 #align principal_ideal_ring.to_unique_factorization_monoid PrincipalIdealRing.to_uniqueFactorizationMonoid
 
 end
@@ -354,30 +427,23 @@ section
 
 open Ideal
 
-variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
+variable [CommRing R] [IsDomain R]
+
+section Bezout
+variable [IsBezout R]
 
 section GCD
 variable [GCDMonoid R]
 
-theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) := by
-  obtain ⟨d, hd⟩ := IsPrincipalIdealRing.principal (span ({x, y} : Set R))
-  rw [submodule_span_eq] at hd
-  rw [hd]
-  suffices Associated d (gcd x y) by
-    obtain ⟨D, HD⟩ := this
-    rw [← HD]
-    exact span_singleton_mul_right_unit D.isUnit _
-  apply associated_of_dvd_dvd
-  · rw [dvd_gcd_iff]
-    constructor <;> rw [← Ideal.mem_span_singleton, ← hd, Ideal.mem_span_pair]
-    · use 1, 0
-      rw [one_mul, zero_mul, add_zero]
-    · use 0, 1
-      rw [one_mul, zero_mul, zero_add]
-  · obtain ⟨r, s, rfl⟩ : ∃ r s, r * x + s * y = d := by
-      rw [← Ideal.mem_span_pair, hd, Ideal.mem_span_singleton]
-    apply dvd_add <;> apply dvd_mul_of_dvd_right
-    exacts [gcd_dvd_left x y, gcd_dvd_right x y]
+theorem IsBezout.span_gcd_eq_span_gcd (x y : R) :
+    span {GCDMonoid.gcd x y} = span {IsBezout.gcd x y} := by
+  rw [Ideal.span_singleton_eq_span_singleton]
+  exact associated_of_dvd_dvd
+    (IsBezout.dvd_gcd (GCDMonoid.gcd_dvd_left _ _) <| GCDMonoid.gcd_dvd_right _ _)
+    (GCDMonoid.dvd_gcd (IsBezout.gcd_dvd_left _ _) <| IsBezout.gcd_dvd_right _ _)
+
+theorem span_gcd (x y : R) : span {gcd x y} = span {x, y} := by
+  rw [← IsBezout.span_gcd, IsBezout.span_gcd_eq_span_gcd]
 #align span_gcd span_gcd
 
 theorem gcd_dvd_iff_exists (a b : R) {z} : gcd a b ∣ z ↔ ∃ x y, z = a * x + b * y := by
@@ -396,62 +462,22 @@ theorem gcd_isUnit_iff (x y : R) : IsUnit (gcd x y) ↔ IsCoprime x y := by
 
 end GCD
 
--- this should be proved for UFDs surely?
 theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
-    (H : ∀ z ∈ nonunits R, z ≠ 0 → z ∣ x → ¬z ∣ y) : IsCoprime x y := by
-  letI := UniqueFactorizationMonoid.toGCDMonoid R
-  rw [← gcd_isUnit_iff]
-  by_contra h
-  refine' H _ h _ (gcd_dvd_left _ _) (gcd_dvd_right _ _)
-  rwa [Ne, gcd_eq_zero_iff]
+    (H : ∀ z ∈ nonunits R, z ≠ 0 → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
+  (isRelPrime_of_no_nonunits_factors nonzero H).isCoprime
 #align is_coprime_of_dvd isCoprime_of_dvd
 
--- this should be proved for UFDs surely?
-theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y := by
-  refine' or_iff_not_imp_left.2 fun h' => _
-  apply isCoprime_of_dvd
-  · rintro ⟨rfl, rfl⟩
-    simp at h
-  · rintro z nu - ⟨w, rfl⟩ dy
-    refine' h' (dvd_trans _ dy)
-    simpa using mul_dvd_mul_left z (isUnit_iff_dvd_one.1 <| (of_irreducible_mul h).resolve_left nu)
+theorem dvd_or_coprime (x y : R) (h : Irreducible x) : x ∣ y ∨ IsCoprime x y :=
+  h.dvd_or_isRelPrime.imp_right IsRelPrime.isCoprime
 #align dvd_or_coprime dvd_or_coprime
 
-theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
-    (H : ∀ z : R, Irreducible z → z ∣ x → ¬z ∣ y) : IsCoprime x y := by
-  apply isCoprime_of_dvd x y nonzero
-  intro z znu znz zx zy
-  obtain ⟨i, h1, h2⟩ := WfDvdMonoid.exists_irreducible_factor znu znz
-  apply H i h1 <;>
-    · apply dvd_trans h2
-      assumption
-#align is_coprime_of_irreducible_dvd isCoprime_of_irreducible_dvd
-
-theorem isCoprime_of_prime_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
-    (H : ∀ z : R, Prime z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
-  isCoprime_of_irreducible_dvd nonzero fun z zi ↦
-    H z (PrincipalIdealRing.irreducible_iff_prime.1 zi)
-#align is_coprime_of_prime_dvd isCoprime_of_prime_dvd
-
-theorem Irreducible.coprime_iff_not_dvd {p n : R} (pp : Irreducible p) :
-    IsCoprime p n ↔ ¬p ∣ n := by
-  constructor
-  · intro co H
-    apply pp.not_unit
-    rw [isUnit_iff_dvd_one]
-    apply IsCoprime.dvd_of_dvd_mul_left co
-    rw [mul_one n]
-    exact H
-  · intro nd
-    apply isCoprime_of_irreducible_dvd
-    · rintro ⟨hp, -⟩
-      exact pp.ne_zero hp
-    rintro z zi zp zn
-    exact nd ((zi.associated_of_dvd pp zp).symm.dvd.trans zn)
+/-- See also `Irreducible.isRelPrime_iff_not_dvd`. -/
+theorem Irreducible.coprime_iff_not_dvd {p n : R} (hp : Irreducible p) :
+    IsCoprime p n ↔ ¬p ∣ n := by rw [← isRelPrime_iff_isCoprime, hp.isRelPrime_iff_not_dvd]
 #align irreducible.coprime_iff_not_dvd Irreducible.coprime_iff_not_dvd
 
-theorem Prime.coprime_iff_not_dvd {p n : R} (pp : Prime p) : IsCoprime p n ↔ ¬p ∣ n :=
-  pp.irreducible.coprime_iff_not_dvd
+theorem Prime.coprime_iff_not_dvd {p n : R} (hp : Prime p) : IsCoprime p n ↔ ¬p ∣ n :=
+  hp.irreducible.coprime_iff_not_dvd
 #align prime.coprime_iff_not_dvd Prime.coprime_iff_not_dvd
 
 /-- See also `Irreducible.coprime_iff_not_dvd'`. -/
@@ -469,11 +495,26 @@ theorem Irreducible.coprime_or_dvd {p : R} (hp : Irreducible p) (i : R) : IsCopr
 #align irreducible.coprime_or_dvd Irreducible.coprime_or_dvd
 
 theorem exists_associated_pow_of_mul_eq_pow' {a b c : R} (hab : IsCoprime a b) {k : ℕ}
-    (h : a * b = c ^ k) : ∃ d : R, Associated (d ^ k) a :=
-  letI := UniqueFactorizationMonoid.toGCDMonoid R
-  exists_associated_pow_of_mul_eq_pow ((gcd_isUnit_iff _ _).mpr hab) h
+    (h : a * b = c ^ k) : ∃ d : R, Associated (d ^ k) a := by
+  classical
+  letI := IsBezout.toGCDDomain R
+  exact exists_associated_pow_of_mul_eq_pow ((gcd_isUnit_iff _ _).mpr hab) h
 #align exists_associated_pow_of_mul_eq_pow' exists_associated_pow_of_mul_eq_pow'
 
+end Bezout
+
+variable [IsPrincipalIdealRing R]
+
+theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
+    (H : ∀ z : R, Irreducible z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
+  (WfDvdMonoid.isRelPrime_of_no_irreducible_factors nonzero H).isCoprime
+#align is_coprime_of_irreducible_dvd isCoprime_of_irreducible_dvd
+
+theorem isCoprime_of_prime_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
+    (H : ∀ z : R, Prime z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
+  isCoprime_of_irreducible_dvd nonzero fun z zi ↦ H z zi.prime
+#align is_coprime_of_prime_dvd isCoprime_of_prime_dvd
+
 end
 
 section PrincipalOfPrime

@@ -454,6 +454,7 @@ theorem Prime.coprime_iff_not_dvd {p n : R} (pp : Prime p) : IsCoprime p n ↔ 
   pp.irreducible.coprime_iff_not_dvd
 #align prime.coprime_iff_not_dvd Prime.coprime_iff_not_dvd
 
+/-- See also `Irreducible.coprime_iff_not_dvd'`. -/
 theorem Irreducible.dvd_iff_not_coprime {p n : R} (hp : Irreducible p) : p ∣ n ↔ ¬IsCoprime p n :=
   iff_not_comm.2 hp.coprime_iff_not_dvd
 #align irreducible.dvd_iff_not_coprime Irreducible.dvd_iff_not_coprime

In fact this theorem admits a proof without using any lemmas introduced in #10170.

For this I had to remove some redundant [GCDMonoid R] assumptions in RingTheory/PrincipalIdealDomain.lean.

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

@@ -354,7 +354,10 @@ section
 
 open Ideal
 
-variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] [GCDMonoid R]
+variable [CommRing R] [IsDomain R] [IsPrincipalIdealRing R]
+
+section GCD
+variable [GCDMonoid R]
 
 theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) := by
   obtain ⟨d, hd⟩ := IsPrincipalIdealRing.principal (span ({x, y} : Set R))
@@ -391,9 +394,12 @@ theorem gcd_isUnit_iff (x y : R) : IsUnit (gcd x y) ↔ IsCoprime x y := by
   rw [IsCoprime, ← Ideal.mem_span_pair, ← span_gcd, ← span_singleton_eq_top, eq_top_iff_one]
 #align gcd_is_unit_iff gcd_isUnit_iff
 
+end GCD
+
 -- this should be proved for UFDs surely?
 theorem isCoprime_of_dvd (x y : R) (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z ∈ nonunits R, z ≠ 0 → z ∣ x → ¬z ∣ y) : IsCoprime x y := by
+  letI := UniqueFactorizationMonoid.toGCDMonoid R
   rw [← gcd_isUnit_iff]
   by_contra h
   refine' H _ h _ (gcd_dvd_left _ _) (gcd_dvd_right _ _)
@@ -423,7 +429,8 @@ theorem isCoprime_of_irreducible_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
 
 theorem isCoprime_of_prime_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
     (H : ∀ z : R, Prime z → z ∣ x → ¬z ∣ y) : IsCoprime x y :=
-  isCoprime_of_irreducible_dvd nonzero fun z zi => H z <| GCDMonoid.prime_of_irreducible zi
+  isCoprime_of_irreducible_dvd nonzero fun z zi ↦
+    H z (PrincipalIdealRing.irreducible_iff_prime.1 zi)
 #align is_coprime_of_prime_dvd isCoprime_of_prime_dvd
 
 theorem Irreducible.coprime_iff_not_dvd {p n : R} (pp : Irreducible p) :
@@ -462,6 +469,7 @@ theorem Irreducible.coprime_or_dvd {p : R} (hp : Irreducible p) (i : R) : IsCopr
 
 theorem exists_associated_pow_of_mul_eq_pow' {a b c : R} (hab : IsCoprime a b) {k : ℕ}
     (h : a * b = c ^ k) : ∃ d : R, Associated (d ^ k) a :=
+  letI := UniqueFactorizationMonoid.toGCDMonoid R
   exists_associated_pow_of_mul_eq_pow ((gcd_isUnit_iff _ _).mpr hab) h
 #align exists_associated_pow_of_mul_eq_pow' exists_associated_pow_of_mul_eq_pow'
 

The main result is Module.End.isSemisimple_of_squarefree_aeval_eq_zero

@@ -103,6 +103,7 @@ theorem span_singleton_generator (S : Submodule R M) [S.IsPrincipal] : span R {g
   Eq.symm (Classical.choose_spec (principal S))
 #align submodule.is_principal.span_singleton_generator Submodule.IsPrincipal.span_singleton_generator
 
+@[simp]
 theorem _root_.Ideal.span_singleton_generator (I : Ideal R) [I.IsPrincipal] :
     Ideal.span ({generator I} : Set R) = I :=
   Eq.symm (Classical.choose_spec (principal I))
@@ -129,6 +130,11 @@ section CommRing
 
 variable [CommRing R] [Module R M]
 
+theorem associated_generator_span_self [IsPrincipalIdealRing R] [IsDomain R] (r : R) :
+    Associated (generator <| Ideal.span {r}) r := by
+  rw [← Ideal.span_singleton_eq_span_singleton]
+  exact Ideal.span_singleton_generator _
+
 theorem mem_iff_generator_dvd (S : Ideal R) [S.IsPrincipal] {x : R} : x ∈ S ↔ generator S ∣ x :=
   (mem_iff_eq_smul_generator S).trans (exists_congr fun a => by simp only [mul_comm, smul_eq_mul])
 #align submodule.is_principal.mem_iff_generator_dvd Submodule.IsPrincipal.mem_iff_generator_dvd

• @[mk_iff] class MyPred now generates myPred_iff, not MyPred_iff
• add Lean.Name.decapitalize
• fix indentation and a few typos in the docs/comments.

Partially addresses issue #9129

@@ -69,7 +69,7 @@ instance top_isPrincipal : (⊤ : Submodule R R).IsPrincipal :=
 variable (R)
 
 /-- A ring is a principal ideal ring if all (left) ideals are principal. -/
-@[mk_iff isPrincipalIdealRing_iff]
+@[mk_iff]
 class IsPrincipalIdealRing (R : Type u) [Ring R] : Prop where
   principal : ∀ S : Ideal R, S.IsPrincipal
 #align is_principal_ideal_ring IsPrincipalIdealRing

This converts usages of the pattern

cases h
case inl h' => ...
case inr h' => ...

which derive from mathported code, to the "structured cases" syntax:

cases h with
| inl h' => ...
| inr h' => ...

The case where the subgoals are handled with · instead of case is more contentious (and much more numerous) so I left those alone. This pattern also appears with cases', induction, induction', and rcases. Furthermore, there is a similar transformation for by_cases:

by_cases h : cond
case pos => ...
case neg => ...

is replaced by:

if h : cond then
  ...
else
  ...

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

@@ -173,12 +173,12 @@ theorem to_maximal_ideal [CommRing R] [IsDomain R] [IsPrincipalIdealRing R] {S :
     ⟨(ne_top_iff_one S).1 hpi.1, by
       intro T x hST hxS hxT
       cases' (mem_iff_generator_dvd _).1 (hST <| generator_mem S) with z hz
-      cases hpi.mem_or_mem (show generator T * z ∈ S from hz ▸ generator_mem S)
-      case inl h =>
+      cases hpi.mem_or_mem (show generator T * z ∈ S from hz ▸ generator_mem S) with
+      | inl h =>
         have hTS : T ≤ S
         rwa [← T.span_singleton_generator, Ideal.span_le, singleton_subset_iff]
         exact (hxS <| hTS hxT).elim
-      case inr h =>
+      | inr h =>
         cases' (mem_iff_generator_dvd _).1 h with y hy
         have : generator S ≠ 0 := mt (eq_bot_iff_generator_eq_zero _).2 hS
         rw [← mul_one (generator S), hy, mul_left_comm, mul_right_inj' this] at hz

Replace rcases( with rcases (. Same thing for convert( and congrm(. No other change.

@@ -290,7 +290,7 @@ theorem ne_zero_of_mem_factors {R : Type v} [CommRing R] [IsDomain R] [IsPrincip
 
 theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R} (ha : a ≠ 0)
     (hfac : ∀ b ∈ factors a, b ∈ s) (hunit : ∀ c : Rˣ, (c : R) ∈ s) : a ∈ s := by
-  rcases(factors_spec a ha).2 with ⟨c, hc⟩
+  rcases (factors_spec a ha).2 with ⟨c, hc⟩
   rw [← hc]
   exact mul_mem (multiset_prod_mem _ hfac) (hunit _)
 #align principal_ideal_ring.mem_submonoid_of_factors_subset_of_units_subset PrincipalIdealRing.mem_submonoid_of_factors_subset_of_units_subset

@@ -199,8 +199,8 @@ theorem mod_mem_iff {S : Ideal R} {x y : R} (hy : y ∈ S) : x % y ∈ S ↔ x 
 #align mod_mem_iff mod_mem_iff
 
 -- see Note [lower instance priority]
-instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincipalIdealRing R
-    where principal S :=
+instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincipalIdealRing R where
+  principal S :=
     ⟨if h : { x : R | x ∈ S ∧ x ≠ 0 }.Nonempty then
         have wf : WellFounded (EuclideanDomain.r : R → R → Prop) := EuclideanDomain.r_wellFounded
         have hmin : WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h ∈ S ∧

Search&replace MulZeroClass.mul_zero -> mul_zero, MulZeroClass.zero_mul -> zero_mul.

These were introduced by Mathport, as the full name of mul_zero is actually MulZeroClass.mul_zero (it's exported with the short name).

@@ -362,9 +362,9 @@ theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
   · rw [dvd_gcd_iff]
     constructor <;> rw [← Ideal.mem_span_singleton, ← hd, Ideal.mem_span_pair]
     · use 1, 0
-      rw [one_mul, MulZeroClass.zero_mul, add_zero]
+      rw [one_mul, zero_mul, add_zero]
     · use 0, 1
-      rw [one_mul, MulZeroClass.zero_mul, zero_add]
+      rw [one_mul, zero_mul, zero_add]
   · obtain ⟨r, s, rfl⟩ : ∃ r s, r * x + s * y = d := by
       rw [← Ideal.mem_span_pair, hd, Ideal.mem_span_singleton]
     apply dvd_add <;> apply dvd_mul_of_dvd_right

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

This has nice performance benefits.

@@ -228,7 +228,7 @@ instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincip
 
 end
 
-theorem IsField.isPrincipalIdealRing {R : Type _} [CommRing R] (h : IsField R) :
+theorem IsField.isPrincipalIdealRing {R : Type*} [CommRing R] (h : IsField R) :
     IsPrincipalIdealRing R :=
   @EuclideanDomain.to_principal_ideal_domain R (@Field.toEuclideanDomain R h.toField)
 #align is_field.is_principal_ideal_ring IsField.isPrincipalIdealRing
@@ -297,7 +297,7 @@ theorem mem_submonoid_of_factors_subset_of_units_subset (s : Submonoid R) {a : R
 
 /-- If a `RingHom` maps all units and all factors of an element `a` into a submonoid `s`, then it
 also maps `a` into that submonoid. -/
-theorem ringHom_mem_submonoid_of_factors_subset_of_units_subset {R S : Type _} [CommRing R]
+theorem ringHom_mem_submonoid_of_factors_subset_of_units_subset {R S : Type*} [CommRing R]
     [IsDomain R] [IsPrincipalIdealRing R] [Semiring S] (f : R →+* S) (s : Submonoid S) (a : R)
     (ha : a ≠ 0) (h : ∀ b ∈ factors a, f b ∈ s) (hf : ∀ c : Rˣ, f c ∈ s) : f a ∈ s :=
   mem_submonoid_of_factors_subset_of_units_subset (s.comap f.toMonoidHom) ha h hf
@@ -318,7 +318,7 @@ section Surjective
 
 open Submodule
 
-variable {S N : Type _} [Ring R] [AddCommGroup M] [AddCommGroup N] [Ring S]
+variable {S N : Type*} [Ring R] [AddCommGroup M] [AddCommGroup N] [Ring S]
 
 variable [Module R M] [Module R N]
 

• depends on: #5966

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

@@ -2,15 +2,12 @@
 Copyright (c) 2018 Chris Hughes. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chris Hughes, Morenikeji Neri
-
-! This file was ported from Lean 3 source module ring_theory.principal_ideal_domain
-! leanprover-community/mathlib commit 6010cf523816335f7bae7f8584cb2edaace73940
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Algebra.EuclideanDomain.Instances
 import Mathlib.RingTheory.UniqueFactorizationDomain
 
+#align_import ring_theory.principal_ideal_domain from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
+
 /-!
 # Principal ideal rings and principal ideal domains
 

Too often tempted to change these during other PRs, so doing a mass edit here.

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -371,7 +371,7 @@ theorem span_gcd (x y : R) : span ({gcd x y} : Set R) = span ({x, y} : Set R) :=
   · obtain ⟨r, s, rfl⟩ : ∃ r s, r * x + s * y = d := by
       rw [← Ideal.mem_span_pair, hd, Ideal.mem_span_singleton]
     apply dvd_add <;> apply dvd_mul_of_dvd_right
-    exacts[gcd_dvd_left x y, gcd_dvd_right x y]
+    exacts [gcd_dvd_left x y, gcd_dvd_right x y]
 #align span_gcd span_gcd
 
 theorem gcd_dvd_iff_exists (a b : R) {z} : gcd a b ∣ z ↔ ∃ x y, z = a * x + b * y := by

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

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

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

@@ -65,14 +65,12 @@ instance bot_isPrincipal : (⊥ : Submodule R M).IsPrincipal :=
   ⟨⟨0, by simp⟩⟩
 #align bot_is_principal bot_isPrincipal
 
-set_option synthInstance.etaExperiment true in
 instance top_isPrincipal : (⊤ : Submodule R R).IsPrincipal :=
   ⟨⟨1, Ideal.span_singleton_one.symm⟩⟩
 #align top_is_principal top_isPrincipal
 
 variable (R)
 
-set_option synthInstance.etaExperiment true in
 /-- A ring is a principal ideal ring if all (left) ideals are principal. -/
 @[mk_iff isPrincipalIdealRing_iff]
 class IsPrincipalIdealRing (R : Type u) [Ring R] : Prop where
@@ -81,7 +79,6 @@ class IsPrincipalIdealRing (R : Type u) [Ring R] : Prop where
 
 attribute [instance] IsPrincipalIdealRing.principal
 
-set_option synthInstance.etaExperiment true in
 instance (priority := 100) DivisionRing.isPrincipalIdealRing (K : Type u) [DivisionRing K] :
     IsPrincipalIdealRing K where
   principal S := by
@@ -109,7 +106,6 @@ theorem span_singleton_generator (S : Submodule R M) [S.IsPrincipal] : span R {g
   Eq.symm (Classical.choose_spec (principal S))
 #align submodule.is_principal.span_singleton_generator Submodule.IsPrincipal.span_singleton_generator
 
-set_option synthInstance.etaExperiment true in
 theorem _root_.Ideal.span_singleton_generator (I : Ideal R) [I.IsPrincipal] :
     Ideal.span ({generator I} : Set R) = I :=
   Eq.symm (Classical.choose_spec (principal I))
@@ -136,12 +132,10 @@ section CommRing
 
 variable [CommRing R] [Module R M]
 
-set_option synthInstance.etaExperiment true in
 theorem mem_iff_generator_dvd (S : Ideal R) [S.IsPrincipal] {x : R} : x ∈ S ↔ generator S ∣ x :=
   (mem_iff_eq_smul_generator S).trans (exists_congr fun a => by simp only [mul_comm, smul_eq_mul])
 #align submodule.is_principal.mem_iff_generator_dvd Submodule.IsPrincipal.mem_iff_generator_dvd
 
-set_option synthInstance.etaExperiment true in
 theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_prime : S.IsPrime]
     (ne_bot : S ≠ ⊥) : Prime (generator S) :=
   ⟨fun h => ne_bot ((eq_bot_iff_generator_eq_zero S).2 h), fun h =>
@@ -150,7 +144,6 @@ theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_p
 #align submodule.is_principal.prime_generator_of_is_prime Submodule.IsPrincipal.prime_generator_of_isPrime
 
 -- Note that the converse may not hold if `ϕ` is not injective.
-set_option synthInstance.etaExperiment true in
 theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
     (hx : x ∈ N) : generator (N.map ϕ) ∣ ϕ x := by
   rw [← mem_iff_generator_dvd, Submodule.mem_map]
@@ -158,7 +151,6 @@ theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.ma
 #align submodule.is_principal.generator_map_dvd_of_mem Submodule.IsPrincipal.generator_map_dvd_of_mem
 
 -- Note that the converse may not hold if `ϕ` is not injective.
-set_option synthInstance.etaExperiment true in
 theorem generator_submoduleImage_dvd_of_mem {N O : Submodule R M} (hNO : N ≤ O) (ϕ : O →ₗ[R] R)
     [(ϕ.submoduleImage N).IsPrincipal] {x : M} (hx : x ∈ N) :
     generator (ϕ.submoduleImage N) ∣ ϕ ⟨x, hNO hx⟩ := by
@@ -174,7 +166,6 @@ namespace IsPrime
 
 open Submodule.IsPrincipal Ideal
 
-set_option synthInstance.etaExperiment true in
 -- TODO -- for a non-ID one could perhaps prove that if p < q are prime then q maximal;
 -- 0 isn't prime in a non-ID PIR but the Krull dimension is still <= 1.
 -- The below result follows from this, but we could also use the below result to
@@ -210,7 +201,6 @@ theorem mod_mem_iff {S : Ideal R} {x y : R} (hy : y ∈ S) : x % y ∈ S ↔ x 
     (mod_eq_sub_mul_div x y).symm ▸ S.sub_mem hx (S.mul_mem_right _ hy)⟩
 #align mod_mem_iff mod_mem_iff
 
-set_option synthInstance.etaExperiment true in
 -- see Note [lower instance priority]
 instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincipalIdealRing R
     where principal S :=
@@ -250,7 +240,6 @@ namespace PrincipalIdealRing
 
 open IsPrincipalIdealRing
 
-set_option synthInstance.etaExperiment true in
 -- see Note [lower instance priority]
 instance (priority := 100) isNoetherianRing [Ring R] [IsPrincipalIdealRing R] :
     IsNoetherianRing R :=
@@ -336,7 +325,6 @@ variable {S N : Type _} [Ring R] [AddCommGroup M] [AddCommGroup N] [Ring S]
 
 variable [Module R M] [Module R N]
 
-set_option synthInstance.etaExperiment true in
 theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjective f)
     (S : Submodule R N) [hI : IsPrincipal (S.comap f)] : IsPrincipal S :=
   ⟨⟨f (IsPrincipal.generator (S.comap f)), by
@@ -344,7 +332,6 @@ theorem Submodule.IsPrincipal.of_comap (f : M →ₗ[R] N) (hf : Function.Surjec
         Submodule.map_comap_eq_of_surjective hf]⟩⟩
 #align submodule.is_principal.of_comap Submodule.IsPrincipal.of_comap
 
-set_option synthInstance.etaExperiment true in
 theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f) (I : Ideal S)
     [hI : IsPrincipal (I.comap f)] : IsPrincipal I :=
   ⟨⟨f (IsPrincipal.generator (I.comap f)), by
@@ -352,7 +339,6 @@ theorem Ideal.IsPrincipal.of_comap (f : R →+* S) (hf : Function.Surjective f)
         Ideal.span_singleton_generator, Ideal.map_comap_of_surjective f hf]⟩⟩
 #align ideal.is_principal.of_comap Ideal.IsPrincipal.of_comap
 
-set_option synthInstance.etaExperiment true in
 /-- The surjective image of a principal ideal ring is again a principal ideal ring. -/
 theorem IsPrincipalIdealRing.of_surjective [IsPrincipalIdealRing R] (f : R →+* S)
     (hf : Function.Surjective f) : IsPrincipalIdealRing S :=
@@ -484,13 +470,11 @@ open Set Ideal
 
 variable (R) [CommRing R]
 
-set_option synthInstance.etaExperiment true in
 /-- `nonPrincipals R` is the set of all ideals of `R` that are not principal ideals. -/
 def nonPrincipals :=
   { I : Ideal R | ¬I.IsPrincipal }
 #align non_principals nonPrincipals
 
-set_option synthInstance.etaExperiment true in
 theorem nonPrincipals_def {I : Ideal R} : I ∈ nonPrincipals R ↔ ¬I.IsPrincipal :=
   Iff.rfl
 #align non_principals_def nonPrincipals_def
@@ -501,7 +485,6 @@ theorem nonPrincipals_eq_empty_iff : nonPrincipals R = ∅ ↔ IsPrincipalIdealR
   simp [Set.eq_empty_iff_forall_not_mem, isPrincipalIdealRing_iff, nonPrincipals_def]
 #align non_principals_eq_empty_iff nonPrincipals_eq_empty_iff
 
-set_option synthInstance.etaExperiment true in
 /-- Any chain in the set of non-principal ideals has an upper bound which is non-principal.
 (Namely, the union of the chain is such an upper bound.)
 -/
@@ -518,7 +501,6 @@ theorem nonPrincipals_zorn (c : Set (Ideal R)) (hs : c ⊆ nonPrincipals R)
   exact hs ⟨⟨x, rfl⟩⟩
 #align non_principals_zorn nonPrincipals_zorn
 
-set_option synthInstance.etaExperiment true in
 /-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
 theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
     IsPrincipalIdealRing R := by

As discussed on Zulip

Renames

• supₛ → sSup
• infₛ → sInf
• supᵢ → iSup
• infᵢ → iInf
• bsupₛ → bsSup
• binfₛ → bsInf
• bsupᵢ → biSup
• binfᵢ → biInf
• csupₛ → csSup
• cinfₛ → csInf
• csupᵢ → ciSup
• cinfᵢ → ciInf
• unionₛ → sUnion
• interₛ → sInter
• unionᵢ → iUnion
• interᵢ → iInter
• bunionₛ → bsUnion
• binterₛ → bsInter
• bunionᵢ → biUnion
• binterᵢ → biInter

Co-authored-by: Parcly Taxel <reddeloostw@gmail.com>

@@ -508,13 +508,13 @@ set_option synthInstance.etaExperiment true in
 theorem nonPrincipals_zorn (c : Set (Ideal R)) (hs : c ⊆ nonPrincipals R)
     (hchain : IsChain (· ≤ ·) c) {K : Ideal R} (hKmem : K ∈ c) :
     ∃ I ∈ nonPrincipals R, ∀ J ∈ c, J ≤ I := by
-  refine' ⟨supₛ c, _, fun J hJ => le_supₛ hJ⟩
+  refine' ⟨sSup c, _, fun J hJ => le_sSup hJ⟩
   rintro ⟨x, hx⟩
-  have hxmem : x ∈ supₛ c := hx.symm ▸ Submodule.mem_span_singleton_self x
-  obtain ⟨J, hJc, hxJ⟩ := (Submodule.mem_supₛ_of_directed ⟨K, hKmem⟩ hchain.directedOn).1 hxmem
-  have hsupₛJ : supₛ c = J := le_antisymm (by simp [hx, Ideal.span_le, hxJ]) (le_supₛ hJc)
+  have hxmem : x ∈ sSup c := hx.symm ▸ Submodule.mem_span_singleton_self x
+  obtain ⟨J, hJc, hxJ⟩ := (Submodule.mem_sSup_of_directed ⟨K, hKmem⟩ hchain.directedOn).1 hxmem
+  have hsSupJ : sSup c = J := le_antisymm (by simp [hx, Ideal.span_le, hxJ]) (le_sSup hJc)
   specialize hs hJc
-  rw [← hsupₛJ, hx, nonPrincipals_def] at hs
+  rw [← hsSupJ, hx, nonPrincipals_def] at hs
   exact hs ⟨⟨x, rfl⟩⟩
 #align non_principals_zorn nonPrincipals_zorn
 

This PR puts, with one exception, every single remaining by that lies all by itself on its own line to the previous line, thus matching the current behaviour of start-port.sh. The exception is when the by begins the second or later argument to a tuple or anonymous constructor; see https://github.com/leanprover-community/mathlib4/pull/3825#discussion_r1186702599.

Essentially this is s/\n *by$/ by/g, but with manual editing to satisfy the linter's max-100-char-line requirement. The Python style linter is also modified to catch these "isolated bys".

@@ -216,36 +216,27 @@ instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincip
     where principal S :=
     ⟨if h : { x : R | x ∈ S ∧ x ≠ 0 }.Nonempty then
         have wf : WellFounded (EuclideanDomain.r : R → R → Prop) := EuclideanDomain.r_wellFounded
-        have hmin :
-          WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h ∈ S ∧
+        have hmin : WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h ∈ S ∧
             WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h ≠ 0 :=
           WellFounded.min_mem wf { x : R | x ∈ S ∧ x ≠ 0 } h
         ⟨WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h,
-          Submodule.ext fun x =>
-            ⟨fun hx =>
-              div_add_mod x (WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h) ▸
-                (Ideal.mem_span_singleton.2 <|
-                  dvd_add (dvd_mul_right _ _) <|
-                    by
-                    have :
-                      x % WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h ∉
-                        { x : R | x ∈ S ∧ x ≠ 0 } :=
-                      fun h₁ => WellFounded.not_lt_min wf _ h h₁ (mod_lt x hmin.2)
-                    have : x % WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h = 0 :=
-                      by
-                      simp only [not_and_or, Set.mem_setOf_eq, not_ne_iff] at this
-                      exact this.neg_resolve_left <| (mod_mem_iff hmin.1).2 hx
-                    simp [*]),
+          Submodule.ext fun x => ⟨fun hx =>
+            div_add_mod x (WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h) ▸
+              (Ideal.mem_span_singleton.2 <| dvd_add (dvd_mul_right _ _) <| by
+                have : x % WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h ∉
+                    { x : R | x ∈ S ∧ x ≠ 0 } :=
+                  fun h₁ => WellFounded.not_lt_min wf _ h h₁ (mod_lt x hmin.2)
+                have : x % WellFounded.min wf { x : R | x ∈ S ∧ x ≠ 0 } h = 0 := by
+                  simp only [not_and_or, Set.mem_setOf_eq, not_ne_iff] at this
+                  exact this.neg_resolve_left <| (mod_mem_iff hmin.1).2 hx
+                simp [*]),
               fun hx =>
-              let ⟨y, hy⟩ := Ideal.mem_span_singleton.1 hx
-              hy.symm ▸ S.mul_mem_right _ hmin.1⟩⟩
-      else
-        ⟨0,
-          Submodule.ext fun a => by
+                let ⟨y, hy⟩ := Ideal.mem_span_singleton.1 hx
+                hy.symm ▸ S.mul_mem_right _ hmin.1⟩⟩
+      else ⟨0, Submodule.ext fun a => by
             rw [← @Submodule.bot_coe R R _ _ _, span_eq, Submodule.mem_bot]
-            exact
-              ⟨fun haS => by_contra fun ha0 => h ⟨a, ⟨haS, ha0⟩⟩, fun h₁ =>
-                h₁.symm ▸ S.zero_mem⟩⟩⟩
+            exact ⟨fun haS => by_contra fun ha0 => h ⟨a, ⟨haS, ha0⟩⟩,
+              fun h₁ => h₁.symm ▸ S.zero_mem⟩⟩⟩
 #align euclidean_domain.to_principal_ideal_domain EuclideanDomain.to_principal_ideal_domain
 
 end
@@ -264,8 +255,7 @@ set_option synthInstance.etaExperiment true in
 instance (priority := 100) isNoetherianRing [Ring R] [IsPrincipalIdealRing R] :
     IsNoetherianRing R :=
   isNoetherianRing_iff.2
-    ⟨fun s : Ideal R =>
-      by
+    ⟨fun s : Ideal R => by
       rcases (IsPrincipalIdealRing.principal s).principal with ⟨a, rfl⟩
       rw [← Finset.coe_singleton]
       exact ⟨{a}, SetLike.coe_injective rfl⟩⟩
@@ -273,8 +263,7 @@ instance (priority := 100) isNoetherianRing [Ring R] [IsPrincipalIdealRing R] :
 
 theorem isMaximal_of_irreducible [CommRing R] [IsPrincipalIdealRing R] {p : R}
     (hp : Irreducible p) : Ideal.IsMaximal (span R ({p} : Set R)) :=
-  ⟨⟨mt Ideal.span_singleton_eq_top.1 hp.1, fun I hI =>
-      by
+  ⟨⟨mt Ideal.span_singleton_eq_top.1 hp.1, fun I hI => by
       rcases principal I with ⟨a, rfl⟩
       erw [Ideal.span_singleton_eq_top]
       rcases Ideal.span_singleton_le_span_singleton.1 (le_of_lt hI) with ⟨b, rfl⟩
@@ -448,8 +437,8 @@ theorem isCoprime_of_prime_dvd {x y : R} (nonzero : ¬(x = 0 ∧ y = 0))
   isCoprime_of_irreducible_dvd nonzero fun z zi => H z <| GCDMonoid.prime_of_irreducible zi
 #align is_coprime_of_prime_dvd isCoprime_of_prime_dvd
 
-theorem Irreducible.coprime_iff_not_dvd {p n : R} (pp : Irreducible p) : IsCoprime p n ↔ ¬p ∣ n :=
-  by
+theorem Irreducible.coprime_iff_not_dvd {p n : R} (pp : Irreducible p) :
+    IsCoprime p n ↔ ¬p ∣ n := by
   constructor
   · intro co H
     apply pp.not_unit
@@ -538,8 +527,7 @@ theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPr
   intro J hJ
   -- We will show a maximal element `I ∈ nonPrincipals R` (which exists by Zorn) is prime.
   obtain ⟨I, Ibad, -, Imax⟩ := zorn_nonempty_partialOrder₀ (nonPrincipals R) nonPrincipals_zorn _ hJ
-  have Imax' : ∀ {J}, I < J → J.IsPrincipal :=
-    by
+  have Imax' : ∀ {J}, I < J → J.IsPrincipal := by
     intro J hJ
     by_contra He
     exact hJ.ne (Imax _ ((nonPrincipals_def R).2 He) hJ.le).symm
@@ -567,8 +555,7 @@ theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPr
     erw [ha, mem_span_singleton'] at hisup this
     obtain ⟨v, rfl⟩ := this
     obtain ⟨u, rfl⟩ := hisup
-    have hucolon : u ∈ I.colon (span {v * a}) :=
-      by
+    have hucolon : u ∈ I.colon (span {v * a}) := by
       rw [Ideal.mem_colon_singleton, mul_comm v, ← mul_assoc]
       exact mul_mem_right _ _ hi
     erw [hb, mem_span_singleton'] at hucolon

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

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

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

@@ -65,12 +65,14 @@ instance bot_isPrincipal : (⊥ : Submodule R M).IsPrincipal :=
   ⟨⟨0, by simp⟩⟩
 #align bot_is_principal bot_isPrincipal
 
+set_option synthInstance.etaExperiment true in
 instance top_isPrincipal : (⊤ : Submodule R R).IsPrincipal :=
   ⟨⟨1, Ideal.span_singleton_one.symm⟩⟩
 #align top_is_principal top_isPrincipal
 
 variable (R)
 
+set_option synthInstance.etaExperiment true in
 /-- A ring is a principal ideal ring if all (left) ideals are principal. -/
 @[mk_iff isPrincipalIdealRing_iff]
 class IsPrincipalIdealRing (R : Type u) [Ring R] : Prop where
@@ -79,6 +81,7 @@ class IsPrincipalIdealRing (R : Type u) [Ring R] : Prop where
 
 attribute [instance] IsPrincipalIdealRing.principal
 
+set_option synthInstance.etaExperiment true in
 instance (priority := 100) DivisionRing.isPrincipalIdealRing (K : Type u) [DivisionRing K] :
     IsPrincipalIdealRing K where
   principal S := by
@@ -106,6 +109,7 @@ theorem span_singleton_generator (S : Submodule R M) [S.IsPrincipal] : span R {g
   Eq.symm (Classical.choose_spec (principal S))
 #align submodule.is_principal.span_singleton_generator Submodule.IsPrincipal.span_singleton_generator
 
+set_option synthInstance.etaExperiment true in
 theorem _root_.Ideal.span_singleton_generator (I : Ideal R) [I.IsPrincipal] :
     Ideal.span ({generator I} : Set R) = I :=
   Eq.symm (Classical.choose_spec (principal I))
@@ -132,10 +136,12 @@ section CommRing
 
 variable [CommRing R] [Module R M]
 
+set_option synthInstance.etaExperiment true in
 theorem mem_iff_generator_dvd (S : Ideal R) [S.IsPrincipal] {x : R} : x ∈ S ↔ generator S ∣ x :=
   (mem_iff_eq_smul_generator S).trans (exists_congr fun a => by simp only [mul_comm, smul_eq_mul])
 #align submodule.is_principal.mem_iff_generator_dvd Submodule.IsPrincipal.mem_iff_generator_dvd
 
+set_option synthInstance.etaExperiment true in
 theorem prime_generator_of_isPrime (S : Ideal R) [Submodule.IsPrincipal S] [is_prime : S.IsPrime]
     (ne_bot : S ≠ ⊥) : Prime (generator S) :=
   ⟨fun h => ne_bot ((eq_bot_iff_generator_eq_zero S).2 h), fun h =>
@@ -168,6 +174,7 @@ namespace IsPrime
 
 open Submodule.IsPrincipal Ideal
 
+set_option synthInstance.etaExperiment true in
 -- TODO -- for a non-ID one could perhaps prove that if p < q are prime then q maximal;
 -- 0 isn't prime in a non-ID PIR but the Krull dimension is still <= 1.
 -- The below result follows from this, but we could also use the below result to
@@ -203,6 +210,7 @@ theorem mod_mem_iff {S : Ideal R} {x y : R} (hy : y ∈ S) : x % y ∈ S ↔ x 
     (mod_eq_sub_mul_div x y).symm ▸ S.sub_mem hx (S.mul_mem_right _ hy)⟩
 #align mod_mem_iff mod_mem_iff
 
+set_option synthInstance.etaExperiment true in
 -- see Note [lower instance priority]
 instance (priority := 100) EuclideanDomain.to_principal_ideal_domain : IsPrincipalIdealRing R
     where principal S :=
@@ -251,13 +259,14 @@ namespace PrincipalIdealRing
 
 open IsPrincipalIdealRing
 
+set_option synthInstance.etaExperiment true in
 -- see Note [lower instance priority]
 instance (priority := 100) isNoetherianRing [Ring R] [IsPrincipalIdealRing R] :
     IsNoetherianRing R :=
   isNoetherianRing_iff.2
     ⟨fun s : Ideal R =>
       by
-      rcases(IsPrincipalIdealRing.principal s).principal with ⟨a, rfl⟩
+      rcases (IsPrincipalIdealRing.principal s).principal with ⟨a, rfl⟩
       rw [← Finset.coe_singleton]
       exact ⟨{a}, SetLike.coe_injective rfl⟩⟩
 #align principal_ideal_ring.is_noetherian_ring PrincipalIdealRing.isNoetherianRing
@@ -486,11 +495,13 @@ open Set Ideal
 
 variable (R) [CommRing R]
 
+set_option synthInstance.etaExperiment true in
 /-- `nonPrincipals R` is the set of all ideals of `R` that are not principal ideals. -/
 def nonPrincipals :=
   { I : Ideal R | ¬I.IsPrincipal }
 #align non_principals nonPrincipals
 
+set_option synthInstance.etaExperiment true in
 theorem nonPrincipals_def {I : Ideal R} : I ∈ nonPrincipals R ↔ ¬I.IsPrincipal :=
   Iff.rfl
 #align non_principals_def nonPrincipals_def
@@ -501,6 +512,7 @@ theorem nonPrincipals_eq_empty_iff : nonPrincipals R = ∅ ↔ IsPrincipalIdealR
   simp [Set.eq_empty_iff_forall_not_mem, isPrincipalIdealRing_iff, nonPrincipals_def]
 #align non_principals_eq_empty_iff nonPrincipals_eq_empty_iff
 
+set_option synthInstance.etaExperiment true in
 /-- Any chain in the set of non-principal ideals has an upper bound which is non-principal.
 (Namely, the union of the chain is such an upper bound.)
 -/
@@ -517,6 +529,7 @@ theorem nonPrincipals_zorn (c : Set (Ideal R)) (hs : c ⊆ nonPrincipals R)
   exact hs ⟨⟨x, rfl⟩⟩
 #align non_principals_zorn nonPrincipals_zorn
 
+set_option synthInstance.etaExperiment true in
 /-- If all prime ideals in a commutative ring are principal, so are all other ideals. -/
 theorem IsPrincipalIdealRing.of_prime (H : ∀ P : Ideal R, P.IsPrime → P.IsPrincipal) :
     IsPrincipalIdealRing R := by