ring_theory.localization.submodule | Mathlib porting status

Changes since the initial port

The following section lists changes to this file in mathlib3 and mathlib4 that occured after the initial port. Most recent changes are shown first. Hovering over a commit will show all commits associated with the same mathlib3 commit.

Changes in mathlib3

1ebb2060

(last sync)

Changes in mathlib3port

mathlib3

mathlib3port

86d1873c

bump to nightly-2023-09-24-06

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

5655435f

Diff

@@ -3,9 +3,9 @@ Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
 -/
-import Mathbin.RingTheory.Localization.FractionRing
-import Mathbin.RingTheory.Localization.Ideal
-import Mathbin.RingTheory.PrincipalIdealDomain
+import RingTheory.Localization.FractionRing
+import RingTheory.Localization.Ideal
+import RingTheory.PrincipalIdealDomain
 
 #align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"86d1873c01a723aba6788f0b9051ae3d23b4c1c3"

86d1873c

bump to nightly-2023-07-20-09

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

9c56d0dd

Diff

@@ -2,16 +2,13 @@
 Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-
-! This file was ported from Lean 3 source module ring_theory.localization.submodule
-! leanprover-community/mathlib commit 86d1873c01a723aba6788f0b9051ae3d23b4c1c3
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.RingTheory.Localization.FractionRing
 import Mathbin.RingTheory.Localization.Ideal
 import Mathbin.RingTheory.PrincipalIdealDomain
 
+#align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"86d1873c01a723aba6788f0b9051ae3d23b4c1c3"
+
 /-!
 # Submodules in localizations of commutative rings

86d1873c

bump to nightly-2023-06-13-21

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

829520ac

Diff

@@ -44,53 +44,71 @@ def coeSubmodule (I : Ideal R) : Submodule R S :=
 #align is_localization.coe_submodule IsLocalization.coeSubmodule
 -/
 
+#print IsLocalization.mem_coeSubmodule /-
 theorem mem_coeSubmodule (I : Ideal R) {x : S} :
     x ∈ coeSubmodule S I ↔ ∃ y : R, y ∈ I ∧ algebraMap R S y = x :=
   Iff.rfl
 #align is_localization.mem_coe_submodule IsLocalization.mem_coeSubmodule
+-/
 
+#print IsLocalization.coeSubmodule_mono /-
 theorem coeSubmodule_mono {I J : Ideal R} (h : I ≤ J) : coeSubmodule S I ≤ coeSubmodule S J :=
   Submodule.map_mono h
 #align is_localization.coe_submodule_mono IsLocalization.coeSubmodule_mono
+-/
 
+#print IsLocalization.coeSubmodule_bot /-
 @[simp]
 theorem coeSubmodule_bot : coeSubmodule S (⊥ : Ideal R) = ⊥ := by
   rw [coe_submodule, Submodule.map_bot]
 #align is_localization.coe_submodule_bot IsLocalization.coeSubmodule_bot
+-/
 
+#print IsLocalization.coeSubmodule_top /-
 @[simp]
 theorem coeSubmodule_top : coeSubmodule S (⊤ : Ideal R) = 1 := by
   rw [coe_submodule, Submodule.map_top, Submodule.one_eq_range]
 #align is_localization.coe_submodule_top IsLocalization.coeSubmodule_top
+-/
 
+#print IsLocalization.coeSubmodule_sup /-
 @[simp]
 theorem coeSubmodule_sup (I J : Ideal R) :
     coeSubmodule S (I ⊔ J) = coeSubmodule S I ⊔ coeSubmodule S J :=
   Submodule.map_sup _ _ _
 #align is_localization.coe_submodule_sup IsLocalization.coeSubmodule_sup
+-/
 
+#print IsLocalization.coeSubmodule_mul /-
 @[simp]
 theorem coeSubmodule_mul (I J : Ideal R) :
     coeSubmodule S (I * J) = coeSubmodule S I * coeSubmodule S J :=
   Submodule.map_mul _ _ (Algebra.ofId R S)
 #align is_localization.coe_submodule_mul IsLocalization.coeSubmodule_mul
+-/
 
+#print IsLocalization.coeSubmodule_fg /-
 theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R) :
     Submodule.FG (coeSubmodule S I) ↔ Submodule.FG I :=
   ⟨Submodule.fg_of_fg_map _ (LinearMap.ker_eq_bot.mpr hS), Submodule.FG.map _⟩
 #align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fg
+-/
 
+#print IsLocalization.coeSubmodule_span /-
 @[simp]
 theorem coeSubmodule_span (s : Set R) :
     coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s) := by
   rw [IsLocalization.coeSubmodule, Ideal.span, Submodule.map_span]; rfl
 #align is_localization.coe_submodule_span IsLocalization.coeSubmodule_span
+-/
 
+#print IsLocalization.coeSubmodule_span_singleton /-
 @[simp]
 theorem coeSubmodule_span_singleton (x : R) :
     coeSubmodule S (Ideal.span {x}) = Submodule.span R {(algebraMap R S) x} := by
   rw [coe_submodule_span, Set.image_singleton]
 #align is_localization.coe_submodule_span_singleton IsLocalization.coeSubmodule_span_singleton
+-/
 
 variable {g : R →+* P}
 
@@ -102,37 +120,44 @@ variable [IsLocalization M S]
 
 section
 
-include M
-
+#print IsLocalization.isNoetherianRing /-
 theorem isNoetherianRing (h : IsNoetherianRing R) : IsNoetherianRing S :=
   by
   rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at h ⊢
   exact OrderEmbedding.wellFounded (IsLocalization.orderEmbedding M S).dual h
 #align is_localization.is_noetherian_ring IsLocalization.isNoetherianRing
+-/
 
 end
 
 variable {S Q M}
 
+#print IsLocalization.coeSubmodule_le_coeSubmodule /-
 @[mono]
 theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal R} :
     coeSubmodule S I ≤ coeSubmodule S J ↔ I ≤ J :=
   Submodule.map_le_map_iff_of_injective (IsLocalization.injective _ h) _ _
 #align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmodule
+-/
 
+#print IsLocalization.coeSubmodule_strictMono /-
 @[mono]
 theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
     StrictMono (coeSubmodule S : Ideal R → Submodule R S) :=
   strictMono_of_le_iff_le fun _ _ => (coeSubmodule_le_coeSubmodule h).symm
 #align is_localization.coe_submodule_strict_mono IsLocalization.coeSubmodule_strictMono
+-/
 
 variable (S) {Q M}
 
+#print IsLocalization.coeSubmodule_injective /-
 theorem coeSubmodule_injective (h : M ≤ nonZeroDivisors R) :
     Function.Injective (coeSubmodule S : Ideal R → Submodule R S) :=
   injective_of_le_imp_le _ fun _ _ => (coeSubmodule_le_coeSubmodule h).mp
 #align is_localization.coe_submodule_injective IsLocalization.coeSubmodule_injective
+-/
 
+#print IsLocalization.coeSubmodule_isPrincipal /-
 theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
     (coeSubmodule S I).IsPrincipal ↔ I.IsPrincipal :=
   by
@@ -144,9 +169,11 @@ theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
   · refine' ⟨⟨algebraMap R S x, _⟩⟩
     rw [hx, Ideal.submodule_span_eq, coe_submodule_span_singleton]
 #align is_localization.coe_submodule_is_principal IsLocalization.coeSubmodule_isPrincipal
+-/
 
 variable {S} (M)
 
+#print IsLocalization.mem_span_iff /-
 theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]
     {x : N} {a : Set N} :
     x ∈ Submodule.span S a ↔ ∃ y ∈ Submodule.span R a, ∃ z : M, x = mk' S 1 z • y :=
@@ -175,7 +202,9 @@ theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [Is
   · rintro ⟨y, hy, z, rfl⟩
     exact Submodule.smul_mem _ _ (Submodule.span_subset_span R S _ hy)
 #align is_localization.mem_span_iff IsLocalization.mem_span_iff
+-/
 
+#print IsLocalization.mem_span_map /-
 theorem mem_span_map {x : S} {a : Set R} :
     x ∈ Ideal.span (algebraMap R S '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z : M, x = mk' S y z :=
   by
@@ -189,6 +218,7 @@ theorem mem_span_map {x : S} {a : Set R} :
     refine' ⟨algebraMap R S y, Submodule.map_mem_span_algebraMap_image _ _ hy, z, _⟩
     rw [hz, smul_eq_mul, mul_comm, mul_mk'_eq_mk'_of_mul, mul_one]
 #align is_localization.mem_span_map IsLocalization.mem_span_map
+-/
 
 end IsLocalization
 
@@ -202,27 +232,35 @@ section CommRing
 
 variable [CommRing K] [Algebra R K] [IsFractionRing R K] [Algebra A K] [IsFractionRing A K]
 
+#print IsFractionRing.coeSubmodule_le_coeSubmodule /-
 @[simp, mono]
 theorem coeSubmodule_le_coeSubmodule {I J : Ideal R} :
     coeSubmodule K I ≤ coeSubmodule K J ↔ I ≤ J :=
   IsLocalization.coeSubmodule_le_coeSubmodule le_rfl
 #align is_fraction_ring.coe_submodule_le_coe_submodule IsFractionRing.coeSubmodule_le_coeSubmodule
+-/
 
+#print IsFractionRing.coeSubmodule_strictMono /-
 @[mono]
 theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submodule R K) :=
   strictMono_of_le_iff_le fun _ _ => coeSubmodule_le_coeSubmodule.symm
 #align is_fraction_ring.coe_submodule_strict_mono IsFractionRing.coeSubmodule_strictMono
+-/
 
 variable (R K)
 
+#print IsFractionRing.coeSubmodule_injective /-
 theorem coeSubmodule_injective : Function.Injective (coeSubmodule K : Ideal R → Submodule R K) :=
   injective_of_le_imp_le _ fun _ _ => coeSubmodule_le_coeSubmodule.mp
 #align is_fraction_ring.coe_submodule_injective IsFractionRing.coeSubmodule_injective
+-/
 
+#print IsFractionRing.coeSubmodule_isPrincipal /-
 @[simp]
 theorem coeSubmodule_isPrincipal {I : Ideal R} : (coeSubmodule K I).IsPrincipal ↔ I.IsPrincipal :=
   IsLocalization.coeSubmodule_isPrincipal _ le_rfl
 #align is_fraction_ring.coe_submodule_is_principal IsFractionRing.coeSubmodule_isPrincipal
+-/
 
 end CommRing

86d1873c

bump to nightly-2023-06-01-16

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

b9c87c70

Diff

@@ -106,7 +106,7 @@ include M
 
 theorem isNoetherianRing (h : IsNoetherianRing R) : IsNoetherianRing S :=
   by
-  rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at h⊢
+  rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at h ⊢
   exact OrderEmbedding.wellFounded (IsLocalization.orderEmbedding M S).dual h
 #align is_localization.is_noetherian_ring IsLocalization.isNoetherianRing

86d1873c

bump to nightly-2023-05-28-06

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

ba5d8b97

Diff

@@ -44,102 +44,48 @@ def coeSubmodule (I : Ideal R) : Submodule R S :=
 #align is_localization.coe_submodule IsLocalization.coeSubmodule
 -/
 
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(CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) y) x)))
-Case conversion may be inaccurate. Consider using '#align is_localization.mem_coe_submodule IsLocalization.mem_coeSubmoduleₓ'. -/
 theorem mem_coeSubmodule (I : Ideal R) {x : S} :
     x ∈ coeSubmodule S I ↔ ∃ y : R, y ∈ I ∧ algebraMap R S y = x :=
   Iff.rfl
 #align is_localization.mem_coe_submodule IsLocalization.mem_coeSubmodule
 
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 theorem coeSubmodule_mono {I J : Ideal R} (h : I ≤ J) : coeSubmodule S I ≤ coeSubmodule S J :=
   Submodule.map_mono h
 #align is_localization.coe_submodule_mono IsLocalization.coeSubmodule_mono
 
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 @[simp]
 theorem coeSubmodule_bot : coeSubmodule S (⊥ : Ideal R) = ⊥ := by
   rw [coe_submodule, Submodule.map_bot]
 #align is_localization.coe_submodule_bot IsLocalization.coeSubmodule_bot
 
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 @[simp]
 theorem coeSubmodule_top : coeSubmodule S (⊤ : Ideal R) = 1 := by
   rw [coe_submodule, Submodule.map_top, Submodule.one_eq_range]
 #align is_localization.coe_submodule_top IsLocalization.coeSubmodule_top
 
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 @[simp]
 theorem coeSubmodule_sup (I J : Ideal R) :
     coeSubmodule S (I ⊔ J) = coeSubmodule S I ⊔ coeSubmodule S J :=
   Submodule.map_sup _ _ _
 #align is_localization.coe_submodule_sup IsLocalization.coeSubmodule_sup
 
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 @[simp]
 theorem coeSubmodule_mul (I J : Ideal R) :
     coeSubmodule S (I * J) = coeSubmodule S I * coeSubmodule S J :=
   Submodule.map_mul _ _ (Algebra.ofId R S)
 #align is_localization.coe_submodule_mul IsLocalization.coeSubmodule_mul
 
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 theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R) :
     Submodule.FG (coeSubmodule S I) ↔ Submodule.FG I :=
   ⟨Submodule.fg_of_fg_map _ (LinearMap.ker_eq_bot.mpr hS), Submodule.FG.map _⟩
 #align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fg
 
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 @[simp]
 theorem coeSubmodule_span (s : Set R) :
     coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s) := by
   rw [IsLocalization.coeSubmodule, Ideal.span, Submodule.map_span]; rfl
 #align is_localization.coe_submodule_span IsLocalization.coeSubmodule_span
 
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 @[simp]
 theorem coeSubmodule_span_singleton (x : R) :
     coeSubmodule S (Ideal.span {x}) = Submodule.span R {(algebraMap R S) x} := by
@@ -158,12 +104,6 @@ section
 
 include M
 
-/- warning: is_localization.is_noetherian_ring -> IsLocalization.isNoetherianRing is a dubious translation:
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 theorem isNoetherianRing (h : IsNoetherianRing R) : IsNoetherianRing S :=
   by
   rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at h⊢
@@ -174,18 +114,12 @@ end
 
 variable {S Q M}
 
-/- warning: is_localization.coe_submodule_le_coe_submodule -> IsLocalization.coeSubmodule_le_coeSubmodule is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmoduleₓ'. -/
 @[mono]
 theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal R} :
     coeSubmodule S I ≤ coeSubmodule S J ↔ I ≤ J :=
   Submodule.map_le_map_iff_of_injective (IsLocalization.injective _ h) _ _
 #align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmodule
 
-/- warning: is_localization.coe_submodule_strict_mono -> IsLocalization.coeSubmodule_strictMono is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_strict_mono IsLocalization.coeSubmodule_strictMonoₓ'. -/
 @[mono]
 theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
     StrictMono (coeSubmodule S : Ideal R → Submodule R S) :=
@@ -194,23 +128,11 @@ theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
 
 variable (S) {Q M}
 
-/- warning: is_localization.coe_submodule_injective -> IsLocalization.coeSubmodule_injective is a dubious translation:
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 theorem coeSubmodule_injective (h : M ≤ nonZeroDivisors R) :
     Function.Injective (coeSubmodule S : Ideal R → Submodule R S) :=
   injective_of_le_imp_le _ fun _ _ => (coeSubmodule_le_coeSubmodule h).mp
 #align is_localization.coe_submodule_injective IsLocalization.coeSubmodule_injective
 
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-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u2, u1} R S (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} S (CommRing.toRing.{u1} S _inst_2)) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I))
-Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_is_principal IsLocalization.coeSubmodule_isPrincipalₓ'. -/
 theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
     (coeSubmodule S I).IsPrincipal ↔ I.IsPrincipal :=
   by
@@ -225,9 +147,6 @@ theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
 
 variable {S} (M)
 
-/- warning: is_localization.mem_span_iff -> IsLocalization.mem_span_iff is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_localization.mem_span_iff IsLocalization.mem_span_iffₓ'. -/
 theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]
     {x : N} {a : Set N} :
     x ∈ Submodule.span S a ↔ ∃ y ∈ Submodule.span R a, ∃ z : M, x = mk' S 1 z • y :=
@@ -257,9 +176,6 @@ theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [Is
     exact Submodule.smul_mem _ _ (Submodule.span_subset_span R S _ hy)
 #align is_localization.mem_span_iff IsLocalization.mem_span_iff
 
-/- warning: is_localization.mem_span_map -> IsLocalization.mem_span_map is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_localization.mem_span_map IsLocalization.mem_span_mapₓ'. -/
 theorem mem_span_map {x : S} {a : Set R} :
     x ∈ Ideal.span (algebraMap R S '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z : M, x = mk' S y z :=
   by
@@ -286,24 +202,12 @@ section CommRing
 
 variable [CommRing K] [Algebra R K] [IsFractionRing R K] [Algebra A K] [IsFractionRing A K]
 
-/- warning: is_fraction_ring.coe_submodule_le_coe_submodule -> IsFractionRing.coeSubmodule_le_coeSubmodule is a dubious translation:
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-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {K : Type.{u2}} [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J)
-but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {J : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 J)) (LE.le.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) I J)
-Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_le_coe_submodule IsFractionRing.coeSubmodule_le_coeSubmoduleₓ'. -/
 @[simp, mono]
 theorem coeSubmodule_le_coeSubmodule {I J : Ideal R} :
     coeSubmodule K I ≤ coeSubmodule K J ↔ I ≤ J :=
   IsLocalization.coeSubmodule_le_coeSubmodule le_rfl
 #align is_fraction_ring.coe_submodule_le_coe_submodule IsFractionRing.coeSubmodule_le_coeSubmodule
 
-/- warning: is_fraction_ring.coe_submodule_strict_mono -> IsFractionRing.coeSubmodule_strictMono is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {K : Type.{u2}} [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7], StrictMono.{u1, u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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))))))) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7)
-but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], StrictMono.{u2, u1} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
-Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_strict_mono IsFractionRing.coeSubmodule_strictMonoₓ'. -/
 @[mono]
 theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submodule R K) :=
   strictMono_of_le_iff_le fun _ _ => coeSubmodule_le_coeSubmodule.symm
@@ -311,22 +215,10 @@ theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submo
 
 variable (R K)
 
-/- warning: is_fraction_ring.coe_submodule_injective -> IsFractionRing.coeSubmodule_injective is a dubious translation:
-lean 3 declaration is
-  forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] (K : Type.{u2}) [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7], Function.Injective.{succ u1, succ u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7)
-but is expected to have type
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-Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_injective IsFractionRing.coeSubmodule_injectiveₓ'. -/
 theorem coeSubmodule_injective : Function.Injective (coeSubmodule K : Ideal R → Submodule R K) :=
   injective_of_le_imp_le _ fun _ _ => coeSubmodule_le_coeSubmodule.mp
 #align is_fraction_ring.coe_submodule_injective IsFractionRing.coeSubmodule_injective
 
-/- warning: is_fraction_ring.coe_submodule_is_principal -> IsFractionRing.coeSubmodule_isPrincipal is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
-  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, Iff (Submodule.IsPrincipal.{u2, u1} R K (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} K (CommRing.toRing.{u1} K _inst_6)) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I)
-Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_is_principal IsFractionRing.coeSubmodule_isPrincipalₓ'. -/
 @[simp]
 theorem coeSubmodule_isPrincipal {I : Ideal R} : (coeSubmodule K I).IsPrincipal ↔ I.IsPrincipal :=
   IsLocalization.coeSubmodule_isPrincipal _ le_rfl

86d1873c

bump to nightly-2023-05-28-04

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

6797a637

Diff

@@ -130,10 +130,8 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_span IsLocalization.coeSubmodule_spanₓ'. -/
 @[simp]
 theorem coeSubmodule_span (s : Set R) :
-    coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s) :=
-  by
-  rw [IsLocalization.coeSubmodule, Ideal.span, Submodule.map_span]
-  rfl
+    coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s) := by
+  rw [IsLocalization.coeSubmodule, Ideal.span, Submodule.map_span]; rfl
 #align is_localization.coe_submodule_span IsLocalization.coeSubmodule_span
 
 /- warning: is_localization.coe_submodule_span_singleton -> IsLocalization.coeSubmodule_span_singleton is a dubious translation:

86d1873c

bump to nightly-2023-05-28-03

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

6c6011cf

Diff

@@ -177,10 +177,7 @@ end
 variable {S Q M}
 
 /- warning: is_localization.coe_submodule_le_coe_submodule -> IsLocalization.coeSubmodule_le_coeSubmodule is a dubious translation:
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-but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R 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(Submodule.completeLattice.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J)) (LE.le.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) I J))
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmoduleₓ'. -/
 @[mono]
 theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal R} :
@@ -189,10 +186,7 @@ theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal
 #align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmodule
 
 /- warning: is_localization.coe_submodule_strict_mono -> IsLocalization.coeSubmodule_strictMono is a dubious translation:
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(Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
-but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R 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(NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_strict_mono IsLocalization.coeSubmodule_strictMonoₓ'. -/
 @[mono]
 theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
@@ -234,10 +228,7 @@ theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
 variable {S} (M)
 
 /- warning: is_localization.mem_span_iff -> IsLocalization.mem_span_iff is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : 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)))))) {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {N : Type.{u3}} [_inst_9 : AddCommGroup.{u3} N] [_inst_10 : Module.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_11 : Module.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_12 : IsScalarTower.{u1, u2, u3} R S N (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))) (SMulZeroClass.toHasSmul.{u2, u3} S N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (SMulWithZero.toSmulZeroClass.{u2, u3} S N (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (MulActionWithZero.toSMulWithZero.{u2, u3} S N (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (Module.toMulActionWithZero.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)))) (SMulZeroClass.toHasSmul.{u1, u3} R N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (SMulWithZero.toSmulZeroClass.{u1, u3} R N (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (MulActionWithZero.toSMulWithZero.{u1, u3} R N (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (Module.toMulActionWithZero.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10))))] {x : N} {a : Set.{u3} N}, Iff (Membership.Mem.{u3, u3} N (Submodule.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) N (Submodule.setLike.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)) x (Submodule.span.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11 a)) (Exists.{succ u3} N (fun (y : N) => Exists.{0} (Membership.Mem.{u3, u3} N (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) N (Submodule.setLike.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10)) y (Submodule.span.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10 a)) (fun (H : Membership.Mem.{u3, u3} N (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) N (Submodule.setLike.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10)) y (Submodule.span.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10 a)) => Exists.{succ u1} (coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) (fun (z : coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) => Eq.{succ u3} N x (SMul.smul.{u2, u3} S N (SMulZeroClass.toHasSmul.{u2, u3} S N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (SMulWithZero.toSmulZeroClass.{u2, u3} S N (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (MulActionWithZero.toSMulWithZero.{u2, u3} S N (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (Module.toMulActionWithZero.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)))) (IsLocalization.mk'.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3 _inst_8 (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) z) y)))))
-but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {N : Type.{u3}} [_inst_9 : AddCommGroup.{u3} N] [_inst_10 : Module.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_11 : Module.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_12 : IsScalarTower.{u2, u1, u3} R S N (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (SMulZeroClass.toSMul.{u1, u3} S N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S N (CommMonoidWithZero.toZero.{u1} S (CommSemiring.toCommMonoidWithZero.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S N (Semiring.toMonoidWithZero.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)))) (SMulZeroClass.toSMul.{u2, u3} R N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R N (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R N (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10))))] {x : N} {a : Set.{u3} N}, Iff (Membership.mem.{u3, u3} N (Submodule.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) (SetLike.instMembership.{u3, u3} (Submodule.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) N (Submodule.setLike.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)) x (Submodule.span.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11 a)) (Exists.{succ u3} N (fun (y : N) => And (Membership.mem.{u3, u3} N (Submodule.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) N (Submodule.setLike.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10)) y (Submodule.span.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10 a)) (Exists.{succ u2} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) x M)) (fun (z : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) x M)) => Eq.{succ u3} N x (HSMul.hSMul.{u1, u3, u3} S N N (instHSMul.{u1, u3} S N (SMulZeroClass.toSMul.{u1, u3} S N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S N (CommMonoidWithZero.toZero.{u1} S (CommSemiring.toCommMonoidWithZero.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S N (Semiring.toMonoidWithZero.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11))))) (IsLocalization.mk'.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3 _inst_8 (OfNat.ofNat.{u2} R 1 (One.toOfNat1.{u2} R (Semiring.toOne.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) z) y)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_localization.mem_span_iff IsLocalization.mem_span_iffₓ'. -/
 theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]
     {x : N} {a : Set N} :
@@ -269,10 +260,7 @@ theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [Is
 #align is_localization.mem_span_iff IsLocalization.mem_span_iff
 
 /- warning: is_localization.mem_span_map -> IsLocalization.mem_span_map is a dubious translation:
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-but is expected to have type
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+<too large>
 Case conversion may be inaccurate. Consider using '#align is_localization.mem_span_map IsLocalization.mem_span_mapₓ'. -/
 theorem mem_span_map {x : S} {a : Set R} :
     x ∈ Ideal.span (algebraMap R S '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z : M, x = mk' S y z :=

86d1873c

bump to nightly-2023-05-19-16

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

a31df521

Diff

@@ -48,7 +48,7 @@ def coeSubmodule (I : Ideal R) : Submodule R S :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) {x : S}, Iff (Membership.Mem.{u2, u2} S (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) S (Submodule.setLike.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) x (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Exists.{succ u1} R (fun (y : R) => And (Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{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))))) y I) (Eq.{succ u2} S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) y) x)))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) {x : S}, Iff (Membership.mem.{u1, u1} S (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) S (Submodule.setLike.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))) x (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) y I) (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) y) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) y) x)))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) {x : S}, Iff (Membership.mem.{u1, u1} S (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) S (Submodule.setLike.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))) x (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) y I) (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) y) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) y) x)))
 Case conversion may be inaccurate. Consider using '#align is_localization.mem_coe_submodule IsLocalization.mem_coeSubmoduleₓ'. -/
 theorem mem_coeSubmodule (I : Ideal R) {x : S} :
     x ∈ coeSubmodule S I ↔ ∃ y : R, y ∈ I ∧ algebraMap R S y = x :=
@@ -115,7 +115,7 @@ theorem coeSubmodule_mul (I J : Ideal R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], (Function.Injective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Iff (Submodule.FG.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.FG.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))], (Function.Injective.{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 (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))), Iff (Submodule.FG.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.FG.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))], (Function.Injective.{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 (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))), Iff (Submodule.FG.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.FG.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fgₓ'. -/
 theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R) :
     Submodule.FG (coeSubmodule S I) ↔ Submodule.FG I :=
@@ -126,7 +126,7 @@ theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R)
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (s : Set.{u1} R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) s)) (Submodule.span.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) s))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] (s : Set.{u2} R), Eq.{succ u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) s)) (Submodule.span.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) s))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] (s : Set.{u2} R), Eq.{succ u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) s)) (Submodule.span.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) s))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_span IsLocalization.coeSubmodule_spanₓ'. -/
 @[simp]
 theorem coeSubmodule_span (s : Set R) :
@@ -140,7 +140,7 @@ theorem coeSubmodule_span (s : Set R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) x))) (Submodule.span.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (Singleton.singleton.{u2, u2} S (Set.{u2} S) (Set.hasSingleton.{u2} S) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) x)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) x))) (Submodule.span.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonAssocRing.toNonUnitalNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Ring.toNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) _inst_2))))) (Algebra.toModule.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toCommSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) _inst_2)) _inst_3) (Singleton.singleton.{u2, u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Set.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (Set.instSingletonSet.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3) x)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) x))) (Submodule.span.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x) (NonAssocRing.toNonUnitalNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x) (Ring.toNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x) (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x) _inst_2))))) (Algebra.toModule.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x) (CommRing.toCommSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x) _inst_2)) _inst_3) (Singleton.singleton.{u2, u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x) (Set.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x)) (Set.instSingletonSet.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) x)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3) x)))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_span_singleton IsLocalization.coeSubmodule_span_singletonₓ'. -/
 @[simp]
 theorem coeSubmodule_span_singleton (x : R) :
@@ -272,7 +272,7 @@ theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [Is
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : 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)))))) {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {x : S} {a : Set.{u1} R}, Iff (Membership.Mem.{u2, u2} S (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (SetLike.hasMem.{u2, u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) S (Submodule.setLike.{u2, u2} S S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Semiring.toModule.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) x (Ideal.span.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) a))) (Exists.{succ u1} R (fun (y : R) => Exists.{0} (Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{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))))) y (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) a)) (fun (H : Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{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))))) y (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) a)) => Exists.{succ u1} (coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) (fun (z : coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) => Eq.{succ u2} S x (IsLocalization.mk'.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3 _inst_8 y z)))))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {x : S} {a : Set.{u2} R}, Iff (Membership.mem.{u1, u1} S (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) x (Ideal.span.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) a))) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) y (Ideal.span.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) a)) (Exists.{succ u2} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) x M)) (fun (z : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) x M)) => Eq.{succ u1} S x (IsLocalization.mk'.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3 _inst_8 y z)))))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {x : S} {a : Set.{u2} R}, Iff (Membership.mem.{u1, u1} S (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) x (Ideal.span.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) a))) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) y (Ideal.span.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) a)) (Exists.{succ u2} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) x M)) (fun (z : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) x M)) => Eq.{succ u1} S x (IsLocalization.mk'.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3 _inst_8 y z)))))
 Case conversion may be inaccurate. Consider using '#align is_localization.mem_span_map IsLocalization.mem_span_mapₓ'. -/
 theorem mem_span_map {x : S} {a : Set R} :
     x ∈ Ideal.span (algebraMap R S '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z : M, x = mk' S y z :=

86d1873c

bump to nightly-2023-05-16-16

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

9cb92e8c

Diff

@@ -57,7 +57,7 @@ theorem mem_coeSubmodule (I : Ideal R) {x : S} :
 
 /- warning: is_localization.coe_submodule_mono -> IsLocalization.coeSubmodule_mono is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : 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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J) -> (LE.le.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : 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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J) -> (LE.le.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
 but is expected to have type
   forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {J : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, (LE.le.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) I J) -> (LE.le.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_mono IsLocalization.coeSubmodule_monoₓ'. -/
@@ -178,7 +178,7 @@ variable {S Q M}
 
 /- warning: is_localization.coe_submodule_le_coe_submodule -> IsLocalization.coeSubmodule_le_coeSubmodule is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (forall {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toHasLe.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (forall {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J))
 but is expected to have type
   forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) -> (forall {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {J : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J)) (LE.le.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) I J))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmoduleₓ'. -/
@@ -190,7 +190,7 @@ theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal
 
 /- warning: is_localization.coe_submodule_strict_mono -> IsLocalization.coeSubmodule_strictMono is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (StrictMono.{u1, u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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))))))) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toHasLe.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (StrictMono.{u1, u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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))))))) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
 but is expected to have type
   forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) -> (StrictMono.{u2, u1} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_strict_mono IsLocalization.coeSubmodule_strictMonoₓ'. -/
@@ -204,7 +204,7 @@ variable (S) {Q M}
 
 /- warning: is_localization.coe_submodule_injective -> IsLocalization.coeSubmodule_injective is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Function.Injective.{succ u1, succ u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toHasLe.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Function.Injective.{succ u1, succ u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
 but is expected to have type
   forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) -> (Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_injective IsLocalization.coeSubmodule_injectiveₓ'. -/
@@ -215,7 +215,7 @@ theorem coeSubmodule_injective (h : M ≤ nonZeroDivisors R) :
 
 /- warning: is_localization.coe_submodule_is_principal -> IsLocalization.coeSubmodule_isPrincipal is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u1, u2} R S (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (LE.le.{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)))))) (Preorder.toHasLe.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u1, u2} R S (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
 but is expected to have type
   forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u2, u1} R S (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} S (CommRing.toRing.{u1} S _inst_2)) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_is_principal IsLocalization.coeSubmodule_isPrincipalₓ'. -/
@@ -302,7 +302,7 @@ variable [CommRing K] [Algebra R K] [IsFractionRing R K] [Algebra A K] [IsFracti
 
 /- warning: is_fraction_ring.coe_submodule_le_coe_submodule -> IsFractionRing.coeSubmodule_le_coeSubmodule is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {K : Type.{u2}} [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {K : Type.{u2}} [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J)
 but is expected to have type
   forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {J : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 J)) (LE.le.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) I J)
 Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_le_coe_submodule IsFractionRing.coeSubmodule_le_coeSubmoduleₓ'. -/

86d1873c

bump to nightly-2023-05-16-10

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

7f411d29

Diff

@@ -113,13 +113,13 @@ theorem coeSubmodule_mul (I J : Ideal R) :
 
 /- warning: is_localization.coe_submodule_fg -> IsLocalization.coeSubmodule_fg is a dubious translation:
 lean 3 declaration is
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], (Function.Injective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Iff (Submodule.Fg.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.Fg.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], (Function.Injective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Iff (Submodule.FG.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.FG.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))], (Function.Injective.{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 (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))), Iff (Submodule.Fg.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.Fg.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))], (Function.Injective.{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 (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))), Iff (Submodule.FG.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.FG.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fgₓ'. -/
 theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R) :
-    Submodule.Fg (coeSubmodule S I) ↔ Submodule.Fg I :=
-  ⟨Submodule.fg_of_fg_map _ (LinearMap.ker_eq_bot.mpr hS), Submodule.Fg.map _⟩
+    Submodule.FG (coeSubmodule S I) ↔ Submodule.FG I :=
+  ⟨Submodule.fg_of_fg_map _ (LinearMap.ker_eq_bot.mpr hS), Submodule.FG.map _⟩
 #align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fg
 
 /- warning: is_localization.coe_submodule_span -> IsLocalization.coeSubmodule_span is a dubious translation:

86d1873c

bump to nightly-2023-05-08-22

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

63e712c6

Diff

@@ -48,7 +48,7 @@ def coeSubmodule (I : Ideal R) : Submodule R S :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) {x : S}, Iff (Membership.Mem.{u2, u2} S (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) S (Submodule.setLike.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) x (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Exists.{succ u1} R (fun (y : R) => And (Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{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))))) y I) (Eq.{succ u2} S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) y) x)))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) {x : S}, Iff (Membership.mem.{u1, u1} S (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) S (Submodule.setLike.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3))) x (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) y I) (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) y) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) y) x)))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) {x : S}, Iff (Membership.mem.{u1, u1} S (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) S (Submodule.setLike.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))) x (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) y I) (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) y) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) y) x)))
 Case conversion may be inaccurate. Consider using '#align is_localization.mem_coe_submodule IsLocalization.mem_coeSubmoduleₓ'. -/
 theorem mem_coeSubmodule (I : Ideal R) {x : S} :
     x ∈ coeSubmodule S I ↔ ∃ y : R, y ∈ I ∧ algebraMap R S y = x :=
@@ -59,7 +59,7 @@ theorem mem_coeSubmodule (I : Ideal R) {x : S} :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : 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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J) -> (LE.le.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J) -> (LE.le.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {J : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, (LE.le.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) I J) -> (LE.le.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_mono IsLocalization.coeSubmodule_monoₓ'. -/
 theorem coeSubmodule_mono {I J : Ideal R} (h : I ≤ J) : coeSubmodule S I ≤ coeSubmodule S J :=
   Submodule.map_mono h
@@ -69,7 +69,7 @@ theorem coeSubmodule_mono {I J : Ideal R} (h : I ≤ J) : coeSubmodule S I ≤ c
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (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)))))) (Bot.bot.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.hasBot.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (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)))))) (Bot.bot.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.instBotSubmodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (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)))))) (Bot.bot.{u2} (Submodule.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3)) (Submodule.instBotSubmodule.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3)))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_bot IsLocalization.coeSubmodule_botₓ'. -/
 @[simp]
 theorem coeSubmodule_bot : coeSubmodule S (⊥ : Ideal R) = ⊥ := by
@@ -80,7 +80,7 @@ theorem coeSubmodule_bot : coeSubmodule S (⊥ : Ideal R) = ⊥ := by
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Top.top.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasTop.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (OfNat.ofNat.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) 1 (OfNat.mk.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) 1 (One.one.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.one.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Top.top.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.instTopSubmodule.{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)))))) (OfNat.ofNat.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) 1 (One.toOfNat1.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.one.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Top.top.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.instTopSubmodule.{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)))))) (OfNat.ofNat.{u2} (Submodule.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3)) 1 (One.toOfNat1.{u2} (Submodule.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3)) (Submodule.one.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3)))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_top IsLocalization.coeSubmodule_topₓ'. -/
 @[simp]
 theorem coeSubmodule_top : coeSubmodule S (⊤ : Ideal R) = 1 := by
@@ -91,7 +91,7 @@ theorem coeSubmodule_top : coeSubmodule S (⊤ : Ideal R) = 1 := by
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Sup.sup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SemilatticeSup.toHasSup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (IdemSemiring.toSemilatticeSup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.idemSemiring.{u1, u1} R (CommRing.toCommSemiring.{u1} R _inst_1) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) I J)) (Sup.sup.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (SemilatticeSup.toHasSup.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IdemSemiring.toSemilatticeSup.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.idemSemiring.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Sup.sup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (SemilatticeSup.toSup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (IdemCommSemiring.toSemilatticeSup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.instIdemCommSemiringIdealToSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) I J)) (Sup.sup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IdemCommSemiring.toSemilatticeSup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.instIdemCommSemiringSubmoduleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonAssocSemiringToSemiringToModule.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (J : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))), Eq.{succ u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Sup.sup.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SemilatticeSup.toSup.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (IdemCommSemiring.toSemilatticeSup.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instIdemCommSemiringIdealToSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) I J)) (Sup.sup.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (IdemCommSemiring.toSemilatticeSup.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.instIdemCommSemiringSubmoduleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonAssocSemiringToSemiringToModule.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_sup IsLocalization.coeSubmodule_supₓ'. -/
 @[simp]
 theorem coeSubmodule_sup (I J : Ideal R) :
@@ -103,7 +103,7 @@ theorem coeSubmodule_sup (I J : Ideal R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (HMul.hMul.{u1, u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHMul.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasMul.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) I J)) (HMul.hMul.{u2, u2, u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (instHMul.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.mul.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (HMul.hMul.{u2, u2, u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (instHMul.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.instMulIdealToSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I J)) (HMul.hMul.{u1, u1, u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (instHMul.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.mul.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (J : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))), Eq.{succ u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (HMul.hMul.{u2, u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (instHMul.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Ideal.instMulIdealToSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I J)) (HMul.hMul.{u1, u1, u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (instHMul.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.mul.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_mul IsLocalization.coeSubmodule_mulₓ'. -/
 @[simp]
 theorem coeSubmodule_mul (I J : Ideal R) :
@@ -115,7 +115,7 @@ theorem coeSubmodule_mul (I J : Ideal R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], (Function.Injective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Iff (Submodule.Fg.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.Fg.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))], (Function.Injective.{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 (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))), Iff (Submodule.Fg.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.Fg.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))], (Function.Injective.{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 (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))), Iff (Submodule.Fg.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.Fg.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fgₓ'. -/
 theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R) :
     Submodule.Fg (coeSubmodule S I) ↔ Submodule.Fg I :=
@@ -126,7 +126,7 @@ theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R)
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (s : Set.{u1} R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) s)) (Submodule.span.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) s))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (s : Set.{u2} R), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) s)) (Submodule.span.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) s))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] (s : Set.{u2} R), Eq.{succ u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) s)) (Submodule.span.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) s))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_span IsLocalization.coeSubmodule_spanₓ'. -/
 @[simp]
 theorem coeSubmodule_span (s : Set R) :
@@ -140,7 +140,7 @@ theorem coeSubmodule_span (s : Set R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) x))) (Submodule.span.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (Singleton.singleton.{u2, u2} S (Set.{u2} S) (Set.hasSingleton.{u2} S) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) x)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) x))) (Submodule.span.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonAssocRing.toNonUnitalNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Ring.toNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) _inst_2))))) (Algebra.toModule.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) _inst_2)) _inst_3) (Singleton.singleton.{u2, u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Set.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (Set.instSingletonSet.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) x)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) x))) (Submodule.span.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonAssocRing.toNonUnitalNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Ring.toNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) _inst_2))))) (Algebra.toModule.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toCommSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) _inst_2)) _inst_3) (Singleton.singleton.{u2, u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Set.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (Set.instSingletonSet.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2))))))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_2)) _inst_3) x)))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_span_singleton IsLocalization.coeSubmodule_span_singletonₓ'. -/
 @[simp]
 theorem coeSubmodule_span_singleton (x : R) :
@@ -164,7 +164,7 @@ include M
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : 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)))))) (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (IsNoetherianRing.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) -> (IsNoetherianRing.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (IsNoetherianRing.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) -> (IsNoetherianRing.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (IsNoetherianRing.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) -> (IsNoetherianRing.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))
 Case conversion may be inaccurate. Consider using '#align is_localization.is_noetherian_ring IsLocalization.isNoetherianRingₓ'. -/
 theorem isNoetherianRing (h : IsNoetherianRing R) : IsNoetherianRing S :=
   by
@@ -180,7 +180,7 @@ variable {S Q M}
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (forall {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (forall {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J)) (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) -> (forall {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {J : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J)) (LE.le.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) I J))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmoduleₓ'. -/
 @[mono]
 theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal R} :
@@ -192,7 +192,7 @@ theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (StrictMono.{u1, u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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))))))) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (StrictMono.{u2, u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) -> (StrictMono.{u2, u1} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_strict_mono IsLocalization.coeSubmodule_strictMonoₓ'. -/
 @[mono]
 theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
@@ -206,7 +206,7 @@ variable (S) {Q M}
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Function.Injective.{succ u1, succ u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) -> (Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_injective IsLocalization.coeSubmodule_injectiveₓ'. -/
 theorem coeSubmodule_injective (h : M ≤ nonZeroDivisors R) :
     Function.Injective (coeSubmodule S : Ideal R → Submodule R S) :=
@@ -217,7 +217,7 @@ theorem coeSubmodule_injective (h : M ≤ nonZeroDivisors R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u1, u2} R S (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u2, u1} R S (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} S (CommRing.toRing.{u1} S _inst_2)) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u2, u1} R S (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} S (CommRing.toRing.{u1} S _inst_2)) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_is_principal IsLocalization.coeSubmodule_isPrincipalₓ'. -/
 theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
     (coeSubmodule S I).IsPrincipal ↔ I.IsPrincipal :=
@@ -237,7 +237,7 @@ variable {S} (M)
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : 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)))))) {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {N : Type.{u3}} [_inst_9 : AddCommGroup.{u3} N] [_inst_10 : Module.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_11 : Module.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_12 : IsScalarTower.{u1, u2, u3} R S N (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))) (SMulZeroClass.toHasSmul.{u2, u3} S N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (SMulWithZero.toSmulZeroClass.{u2, u3} S N (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (MulActionWithZero.toSMulWithZero.{u2, u3} S N (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (Module.toMulActionWithZero.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)))) (SMulZeroClass.toHasSmul.{u1, u3} R N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (SMulWithZero.toSmulZeroClass.{u1, u3} R N (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (MulActionWithZero.toSMulWithZero.{u1, u3} R N (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (Module.toMulActionWithZero.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10))))] {x : N} {a : Set.{u3} N}, Iff (Membership.Mem.{u3, u3} N (Submodule.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) N (Submodule.setLike.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)) x (Submodule.span.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11 a)) (Exists.{succ u3} N (fun (y : N) => Exists.{0} (Membership.Mem.{u3, u3} N (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) N (Submodule.setLike.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10)) y (Submodule.span.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10 a)) (fun (H : Membership.Mem.{u3, u3} N (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) N (Submodule.setLike.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10)) y (Submodule.span.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10 a)) => Exists.{succ u1} (coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) (fun (z : coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) => Eq.{succ u3} N x (SMul.smul.{u2, u3} S N (SMulZeroClass.toHasSmul.{u2, u3} S N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (SMulWithZero.toSmulZeroClass.{u2, u3} S N (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (MulActionWithZero.toSMulWithZero.{u2, u3} S N (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (Module.toMulActionWithZero.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)))) (IsLocalization.mk'.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3 _inst_8 (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) z) y)))))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {N : Type.{u3}} [_inst_9 : AddCommGroup.{u3} N] [_inst_10 : Module.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_11 : Module.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_12 : IsScalarTower.{u2, u1, u3} R S N (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) (SMulZeroClass.toSMul.{u1, u3} S N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S N (CommMonoidWithZero.toZero.{u1} S (CommSemiring.toCommMonoidWithZero.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S N (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)))) (SMulZeroClass.toSMul.{u2, u3} R N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R N (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R N (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10))))] {x : N} {a : Set.{u3} N}, Iff (Membership.mem.{u3, u3} N (Submodule.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) (SetLike.instMembership.{u3, u3} (Submodule.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) N (Submodule.setLike.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)) x (Submodule.span.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11 a)) (Exists.{succ u3} N (fun (y : N) => And (Membership.mem.{u3, u3} N (Submodule.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) N (Submodule.setLike.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10)) y (Submodule.span.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10 a)) (Exists.{succ u2} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) x M)) (fun (z : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) x M)) => Eq.{succ u3} N x (HSMul.hSMul.{u1, u3, u3} S N N (instHSMul.{u1, u3} S N (SMulZeroClass.toSMul.{u1, u3} S N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S N (CommMonoidWithZero.toZero.{u1} S (CommSemiring.toCommMonoidWithZero.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S N (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11))))) (IsLocalization.mk'.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3 _inst_8 (OfNat.ofNat.{u2} R 1 (One.toOfNat1.{u2} R (NonAssocRing.toOne.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) z) y)))))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {N : Type.{u3}} [_inst_9 : AddCommGroup.{u3} N] [_inst_10 : Module.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_11 : Module.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_12 : IsScalarTower.{u2, u1, u3} R S N (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3) (SMulZeroClass.toSMul.{u1, u3} S N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S N (CommMonoidWithZero.toZero.{u1} S (CommSemiring.toCommMonoidWithZero.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S N (Semiring.toMonoidWithZero.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)))) (SMulZeroClass.toSMul.{u2, u3} R N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R N (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R N (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10))))] {x : N} {a : Set.{u3} N}, Iff (Membership.mem.{u3, u3} N (Submodule.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) (SetLike.instMembership.{u3, u3} (Submodule.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) N (Submodule.setLike.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)) x (Submodule.span.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11 a)) (Exists.{succ u3} N (fun (y : N) => And (Membership.mem.{u3, u3} N (Submodule.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) N (Submodule.setLike.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10)) y (Submodule.span.{u2, u3} R N (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10 a)) (Exists.{succ u2} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) x M)) (fun (z : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) x M)) => Eq.{succ u3} N x (HSMul.hSMul.{u1, u3, u3} S N N (instHSMul.{u1, u3} S N (SMulZeroClass.toSMul.{u1, u3} S N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S N (CommMonoidWithZero.toZero.{u1} S (CommSemiring.toCommMonoidWithZero.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S N (Semiring.toMonoidWithZero.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u1, u3} S N (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11))))) (IsLocalization.mk'.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3 _inst_8 (OfNat.ofNat.{u2} R 1 (One.toOfNat1.{u2} R (Semiring.toOne.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) z) y)))))
 Case conversion may be inaccurate. Consider using '#align is_localization.mem_span_iff IsLocalization.mem_span_iffₓ'. -/
 theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]
     {x : N} {a : Set N} :
@@ -272,7 +272,7 @@ theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [Is
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : 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)))))) {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {x : S} {a : Set.{u1} R}, Iff (Membership.Mem.{u2, u2} S (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (SetLike.hasMem.{u2, u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) S (Submodule.setLike.{u2, u2} S S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Semiring.toModule.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) x (Ideal.span.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) a))) (Exists.{succ u1} R (fun (y : R) => Exists.{0} (Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{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))))) y (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) a)) (fun (H : Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{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))))) y (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) a)) => Exists.{succ u1} (coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) (fun (z : coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) => Eq.{succ u2} S x (IsLocalization.mk'.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3 _inst_8 y z)))))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {x : S} {a : Set.{u2} R}, Iff (Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) x (Ideal.span.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) a))) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) y (Ideal.span.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) a)) (Exists.{succ u2} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) x M)) (fun (z : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) x M)) => Eq.{succ u1} S x (IsLocalization.mk'.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3 _inst_8 y z)))))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {x : S} {a : Set.{u2} R}, Iff (Membership.mem.{u1, u1} S (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) S (Submodule.setLike.{u1, u1} S S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (Semiring.toModule.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) x (Ideal.span.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_3)) a))) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) y (Ideal.span.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) a)) (Exists.{succ u2} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) x M)) (fun (z : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) x M)) => Eq.{succ u1} S x (IsLocalization.mk'.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3 _inst_8 y z)))))
 Case conversion may be inaccurate. Consider using '#align is_localization.mem_span_map IsLocalization.mem_span_mapₓ'. -/
 theorem mem_span_map {x : S} {a : Set R} :
     x ∈ Ideal.span (algebraMap R S '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z : M, x = mk' S y z :=
@@ -304,7 +304,7 @@ variable [CommRing K] [Algebra R K] [IsFractionRing R K] [Algebra A K] [IsFracti
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {K : Type.{u2}} [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J)
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 J)) (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J)
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))} {J : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 J)) (LE.le.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) I J)
 Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_le_coe_submodule IsFractionRing.coeSubmodule_le_coeSubmoduleₓ'. -/
 @[simp, mono]
 theorem coeSubmodule_le_coeSubmodule {I J : Ideal R} :
@@ -316,7 +316,7 @@ theorem coeSubmodule_le_coeSubmodule {I J : Ideal R} :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {K : Type.{u2}} [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7], StrictMono.{u1, u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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))))))) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7)
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], StrictMono.{u2, u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], StrictMono.{u2, u1} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
 Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_strict_mono IsFractionRing.coeSubmodule_strictMonoₓ'. -/
 @[mono]
 theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submodule R K) :=
@@ -329,7 +329,7 @@ variable (R K)
 lean 3 declaration is
   forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] (K : Type.{u2}) [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7], Function.Injective.{succ u1, succ u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7)
 but is expected to have type
-  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
+  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Submodule.{u2, u1} R K (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
 Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_injective IsFractionRing.coeSubmodule_injectiveₓ'. -/
 theorem coeSubmodule_injective : Function.Injective (coeSubmodule K : Ideal R → Submodule R K) :=
   injective_of_le_imp_le _ fun _ _ => coeSubmodule_le_coeSubmodule.mp
@@ -339,7 +339,7 @@ theorem coeSubmodule_injective : Function.Injective (coeSubmodule K : Ideal R 
 lean 3 declaration is
   forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] (K : Type.{u2}) [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (Submodule.IsPrincipal.{u1, u2} R K (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6)))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 I)) (Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I)
 but is expected to have type
-  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (Submodule.IsPrincipal.{u2, u1} R K (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} K (CommRing.toRing.{u1} K _inst_6)) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I)
+  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, Iff (Submodule.IsPrincipal.{u2, u1} R K (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} K (CommRing.toRing.{u1} K _inst_6)) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} K (CommRing.toCommSemiring.{u1} K _inst_6)) _inst_7) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I)
 Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_is_principal IsFractionRing.coeSubmodule_isPrincipalₓ'. -/
 @[simp]
 theorem coeSubmodule_isPrincipal {I : Ideal R} : (coeSubmodule K I).IsPrincipal ↔ I.IsPrincipal :=

86d1873c

bump to nightly-2023-04-27-16

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

c8f7a684

Diff

@@ -48,7 +48,7 @@ def coeSubmodule (I : Ideal R) : Submodule R S :=
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) {x : S}, Iff (Membership.Mem.{u2, u2} S (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) S (Submodule.setLike.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) x (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Exists.{succ u1} R (fun (y : R) => And (Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{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))))) y I) (Eq.{succ u2} S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) y) x)))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) {x : S}, Iff (Membership.mem.{u1, u1} S (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) S (Submodule.setLike.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3))) x (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) y I) (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) y) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) y) x)))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) {x : S}, Iff (Membership.mem.{u1, u1} S (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (SetLike.instMembership.{u1, u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) S (Submodule.setLike.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3))) x (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) y I) (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) y) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) y) x)))
 Case conversion may be inaccurate. Consider using '#align is_localization.mem_coe_submodule IsLocalization.mem_coeSubmoduleₓ'. -/
 theorem mem_coeSubmodule (I : Ideal R) {x : S} :
     x ∈ coeSubmodule S I ↔ ∃ y : R, y ∈ I ∧ algebraMap R S y = x :=
@@ -59,7 +59,7 @@ theorem mem_coeSubmodule (I : Ideal R) {x : S} :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : 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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J) -> (LE.le.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J) -> (LE.le.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J) -> (LE.le.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_mono IsLocalization.coeSubmodule_monoₓ'. -/
 theorem coeSubmodule_mono {I J : Ideal R} (h : I ≤ J) : coeSubmodule S I ≤ coeSubmodule S J :=
   Submodule.map_mono h
@@ -69,7 +69,7 @@ theorem coeSubmodule_mono {I J : Ideal R} (h : I ≤ J) : coeSubmodule S I ≤ c
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (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)))))) (Bot.bot.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.hasBot.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (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)))))) (Bot.bot.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (Submodule.instBotSubmodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (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)))))) (Bot.bot.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.instBotSubmodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_bot IsLocalization.coeSubmodule_botₓ'. -/
 @[simp]
 theorem coeSubmodule_bot : coeSubmodule S (⊥ : Ideal R) = ⊥ := by
@@ -80,7 +80,7 @@ theorem coeSubmodule_bot : coeSubmodule S (⊥ : Ideal R) = ⊥ := by
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Top.top.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasTop.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (OfNat.ofNat.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) 1 (OfNat.mk.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) 1 (One.one.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.one.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Top.top.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.instTopSubmodule.{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)))))) (OfNat.ofNat.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) 1 (One.toOfNat1.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (Submodule.one.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Top.top.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.instTopSubmodule.{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)))))) (OfNat.ofNat.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) 1 (One.toOfNat1.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.one.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_top IsLocalization.coeSubmodule_topₓ'. -/
 @[simp]
 theorem coeSubmodule_top : coeSubmodule S (⊤ : Ideal R) = 1 := by
@@ -91,7 +91,7 @@ theorem coeSubmodule_top : coeSubmodule S (⊤ : Ideal R) = 1 := by
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Sup.sup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SemilatticeSup.toHasSup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (IdemSemiring.toSemilatticeSup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.idemSemiring.{u1, u1} R (CommRing.toCommSemiring.{u1} R _inst_1) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) I J)) (Sup.sup.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (SemilatticeSup.toHasSup.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IdemSemiring.toSemilatticeSup.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.idemSemiring.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Sup.sup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (SemilatticeSup.toSup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (IdemCommSemiring.toSemilatticeSup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.instIdemCommSemiringIdealToSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) I J)) (Sup.sup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (IdemCommSemiring.toSemilatticeSup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.instIdemCommSemiringSubmoduleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonAssocSemiringToSemiringToModule.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Sup.sup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (SemilatticeSup.toSup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (IdemCommSemiring.toSemilatticeSup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.instIdemCommSemiringIdealToSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) I J)) (Sup.sup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IdemCommSemiring.toSemilatticeSup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.instIdemCommSemiringSubmoduleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonAssocSemiringToSemiringToModule.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_sup IsLocalization.coeSubmodule_supₓ'. -/
 @[simp]
 theorem coeSubmodule_sup (I J : Ideal R) :
@@ -103,7 +103,7 @@ theorem coeSubmodule_sup (I J : Ideal R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (HMul.hMul.{u1, u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHMul.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasMul.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) I J)) (HMul.hMul.{u2, u2, u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (instHMul.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.mul.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (HMul.hMul.{u2, u2, u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (instHMul.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.instMulIdealToSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I J)) (HMul.hMul.{u1, u1, u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (instHMul.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.mul.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (HMul.hMul.{u2, u2, u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (instHMul.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.instMulIdealToSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) I J)) (HMul.hMul.{u1, u1, u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (instHMul.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.mul.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_mul IsLocalization.coeSubmodule_mulₓ'. -/
 @[simp]
 theorem coeSubmodule_mul (I J : Ideal R) :
@@ -115,7 +115,7 @@ theorem coeSubmodule_mul (I J : Ideal R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], (Function.Injective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Iff (Submodule.Fg.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.Fg.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))], (Function.Injective.{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 (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))), Iff (Submodule.Fg.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.Fg.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))], (Function.Injective.{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 (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))), Iff (Submodule.Fg.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.Fg.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fgₓ'. -/
 theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R) :
     Submodule.Fg (coeSubmodule S I) ↔ Submodule.Fg I :=
@@ -126,7 +126,7 @@ theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R)
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (s : Set.{u1} R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) s)) (Submodule.span.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) s))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (s : Set.{u2} R), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) s)) (Submodule.span.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) s))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (s : Set.{u2} R), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) s)) (Submodule.span.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) s))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_span IsLocalization.coeSubmodule_spanₓ'. -/
 @[simp]
 theorem coeSubmodule_span (s : Set R) :
@@ -140,7 +140,7 @@ theorem coeSubmodule_span (s : Set R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) x))) (Submodule.span.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (Singleton.singleton.{u2, u2} S (Set.{u2} S) (Set.hasSingleton.{u2} S) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) x)))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) x))) (Submodule.span.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonAssocRing.toNonUnitalNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Ring.toNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Singleton.singleton.{u2, u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Set.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (Set.instSingletonSet.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) x)))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) x))) (Submodule.span.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonAssocRing.toNonUnitalNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Ring.toNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) _inst_2))))) (Algebra.toModule.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) _inst_2)) _inst_3) (Singleton.singleton.{u2, u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Set.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (Set.instSingletonSet.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) x)))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_span_singleton IsLocalization.coeSubmodule_span_singletonₓ'. -/
 @[simp]
 theorem coeSubmodule_span_singleton (x : R) :
@@ -180,7 +180,7 @@ variable {S Q M}
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (forall {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (forall {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J)) (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (forall {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J)) (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmoduleₓ'. -/
 @[mono]
 theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal R} :
@@ -192,7 +192,7 @@ theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (StrictMono.{u1, u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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))))))) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (StrictMono.{u2, u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (StrictMono.{u2, u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_strict_mono IsLocalization.coeSubmodule_strictMonoₓ'. -/
 @[mono]
 theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
@@ -206,7 +206,7 @@ variable (S) {Q M}
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Function.Injective.{succ u1, succ u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_injective IsLocalization.coeSubmodule_injectiveₓ'. -/
 theorem coeSubmodule_injective (h : M ≤ nonZeroDivisors R) :
     Function.Injective (coeSubmodule S : Ideal R → Submodule R S) :=
@@ -217,7 +217,7 @@ theorem coeSubmodule_injective (h : M ≤ nonZeroDivisors R) :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u1, u2} R S (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u2, u1} R S (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} S (CommRing.toRing.{u1} S _inst_2)) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I))
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u2, u1} R S (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} S (CommRing.toRing.{u1} S _inst_2)) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I))
 Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_is_principal IsLocalization.coeSubmodule_isPrincipalₓ'. -/
 theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
     (coeSubmodule S I).IsPrincipal ↔ I.IsPrincipal :=
@@ -304,7 +304,7 @@ variable [CommRing K] [Algebra R K] [IsFractionRing R K] [Algebra A K] [IsFracti
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {K : Type.{u2}} [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J)
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (Submodule.completeLattice.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 J)) (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J)
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 J)) (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J)
 Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_le_coe_submodule IsFractionRing.coeSubmodule_le_coeSubmoduleₓ'. -/
 @[simp, mono]
 theorem coeSubmodule_le_coeSubmodule {I J : Ideal R} :
@@ -316,7 +316,7 @@ theorem coeSubmodule_le_coeSubmodule {I J : Ideal R} :
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {K : Type.{u2}} [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7], StrictMono.{u1, u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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))))))) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7)
 but is expected to have type
-  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], StrictMono.{u2, u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (Submodule.completeLattice.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], StrictMono.{u2, u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
 Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_strict_mono IsFractionRing.coeSubmodule_strictMonoₓ'. -/
 @[mono]
 theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submodule R K) :=
@@ -329,7 +329,7 @@ variable (R K)
 lean 3 declaration is
   forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] (K : Type.{u2}) [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7], Function.Injective.{succ u1, succ u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7)
 but is expected to have type
-  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
+  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
 Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_injective IsFractionRing.coeSubmodule_injectiveₓ'. -/
 theorem coeSubmodule_injective : Function.Injective (coeSubmodule K : Ideal R → Submodule R K) :=
   injective_of_le_imp_le _ fun _ _ => coeSubmodule_le_coeSubmodule.mp
@@ -339,7 +339,7 @@ theorem coeSubmodule_injective : Function.Injective (coeSubmodule K : Ideal R 
 lean 3 declaration is
   forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] (K : Type.{u2}) [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (Submodule.IsPrincipal.{u1, u2} R K (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6)))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 I)) (Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I)
 but is expected to have type
-  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (Submodule.IsPrincipal.{u2, u1} R K (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} K (CommRing.toRing.{u1} K _inst_6)) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I)
+  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (Submodule.IsPrincipal.{u2, u1} R K (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} K (CommRing.toRing.{u1} K _inst_6)) (Algebra.toModule.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6)) _inst_7) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I)
 Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_is_principal IsFractionRing.coeSubmodule_isPrincipalₓ'. -/
 @[simp]
 theorem coeSubmodule_isPrincipal {I : Ideal R} : (coeSubmodule K I).IsPrincipal ↔ I.IsPrincipal :=

86d1873c

bump to nightly-2023-04-22-03

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

21a90077

Diff

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
 
 ! This file was ported from Lean 3 source module ring_theory.localization.submodule
-! leanprover-community/mathlib commit 1ebb20602a8caef435ce47f6373e1aa40851a177
+! leanprover-community/mathlib commit 86d1873c01a723aba6788f0b9051ae3d23b4c1c3
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.RingTheory.PrincipalIdealDomain
 /-!
 # Submodules in localizations of commutative rings
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 ## Implementation notes
 
 See `src/ring_theory/localization/basic.lean` for a design overview.

1ebb2060

bump to nightly-2023-04-20-00

mathlib commit https://github.com/leanprover-community/mathlib/commit/cd8fafa2fac98e1a67097e8a91ad9901cfde48af

d6678ca6

Diff

@@ -31,6 +31,7 @@ variable [Algebra R S] {P : Type _} [CommRing P]
 
 namespace IsLocalization
 
+#print IsLocalization.coeSubmodule /-
 -- This was previously a `has_coe` instance, but if `S = R` then this will loop.
 -- It could be a `has_coe_t` instance, but we keep it explicit here to avoid slowing down
 -- the rest of the library.
@@ -38,43 +39,92 @@ namespace IsLocalization
 def coeSubmodule (I : Ideal R) : Submodule R S :=
   Submodule.map (Algebra.linearMap R S) I
 #align is_localization.coe_submodule IsLocalization.coeSubmodule
+-/
 
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align is_localization.mem_coe_submodule IsLocalization.mem_coeSubmoduleₓ'. -/
 theorem mem_coeSubmodule (I : Ideal R) {x : S} :
     x ∈ coeSubmodule S I ↔ ∃ y : R, y ∈ I ∧ algebraMap R S y = x :=
   Iff.rfl
 #align is_localization.mem_coe_submodule IsLocalization.mem_coeSubmodule
 
+/- warning: is_localization.coe_submodule_mono -> IsLocalization.coeSubmodule_mono is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : 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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J) -> (LE.le.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J) -> (LE.le.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_mono IsLocalization.coeSubmodule_monoₓ'. -/
 theorem coeSubmodule_mono {I J : Ideal R} (h : I ≤ J) : coeSubmodule S I ≤ coeSubmodule S J :=
   Submodule.map_mono h
 #align is_localization.coe_submodule_mono IsLocalization.coeSubmodule_mono
 
+/- warning: is_localization.coe_submodule_bot -> IsLocalization.coeSubmodule_bot is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (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)))))) (Bot.bot.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.hasBot.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (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)))))) (Bot.bot.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (Submodule.instBotSubmodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)))
+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_bot IsLocalization.coeSubmodule_botₓ'. -/
 @[simp]
 theorem coeSubmodule_bot : coeSubmodule S (⊥ : Ideal R) = ⊥ := by
   rw [coe_submodule, Submodule.map_bot]
 #align is_localization.coe_submodule_bot IsLocalization.coeSubmodule_bot
 
+/- warning: is_localization.coe_submodule_top -> IsLocalization.coeSubmodule_top is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Top.top.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.hasTop.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (OfNat.ofNat.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) 1 (OfNat.mk.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) 1 (One.one.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.one.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Top.top.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.instTopSubmodule.{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)))))) (OfNat.ofNat.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) 1 (One.toOfNat1.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (Submodule.one.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))
+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_top IsLocalization.coeSubmodule_topₓ'. -/
 @[simp]
 theorem coeSubmodule_top : coeSubmodule S (⊤ : Ideal R) = 1 := by
   rw [coe_submodule, Submodule.map_top, Submodule.one_eq_range]
 #align is_localization.coe_submodule_top IsLocalization.coeSubmodule_top
 
+/- warning: is_localization.coe_submodule_sup -> IsLocalization.coeSubmodule_sup is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Sup.sup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SemilatticeSup.toHasSup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (IdemSemiring.toSemilatticeSup.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.idemSemiring.{u1, u1} R (CommRing.toCommSemiring.{u1} R _inst_1) R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) I J)) (Sup.sup.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (SemilatticeSup.toHasSup.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IdemSemiring.toSemilatticeSup.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.idemSemiring.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Sup.sup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (SemilatticeSup.toSup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (IdemCommSemiring.toSemilatticeSup.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Ideal.instIdemCommSemiringIdealToSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) I J)) (Sup.sup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (SemilatticeSup.toSup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (IdemCommSemiring.toSemilatticeSup.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.instIdemCommSemiringSubmoduleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonAssocSemiringToSemiringToModule.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J))
+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_sup IsLocalization.coeSubmodule_supₓ'. -/
 @[simp]
 theorem coeSubmodule_sup (I J : Ideal R) :
     coeSubmodule S (I ⊔ J) = coeSubmodule S I ⊔ coeSubmodule S J :=
   Submodule.map_sup _ _ _
 #align is_localization.coe_submodule_sup IsLocalization.coeSubmodule_sup
 
+/- warning: is_localization.coe_submodule_mul -> IsLocalization.coeSubmodule_mul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (HMul.hMul.{u1, u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (instHMul.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasMul.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) I J)) (HMul.hMul.{u2, u2, u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (instHMul.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.mul.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_mul IsLocalization.coeSubmodule_mulₓ'. -/
 @[simp]
 theorem coeSubmodule_mul (I J : Ideal R) :
     coeSubmodule S (I * J) = coeSubmodule S I * coeSubmodule S J :=
   Submodule.map_mul _ _ (Algebra.ofId R S)
 #align is_localization.coe_submodule_mul IsLocalization.coeSubmodule_mul
 
+/- warning: is_localization.coe_submodule_fg -> IsLocalization.coeSubmodule_fg is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))], (Function.Injective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))) -> (forall (I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), Iff (Submodule.Fg.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.Fg.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fgₓ'. -/
 theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R) :
     Submodule.Fg (coeSubmodule S I) ↔ Submodule.Fg I :=
   ⟨Submodule.fg_of_fg_map _ (LinearMap.ker_eq_bot.mpr hS), Submodule.Fg.map _⟩
 #align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fg
 
+/- warning: is_localization.coe_submodule_span -> IsLocalization.coeSubmodule_span is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (s : Set.{u1} R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) s)) (Submodule.span.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) s))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] (s : Set.{u2} R), Eq.{succ u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) s)) (Submodule.span.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) s))
+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_span IsLocalization.coeSubmodule_spanₓ'. -/
 @[simp]
 theorem coeSubmodule_span (s : Set R) :
     coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s) :=
@@ -83,6 +133,12 @@ theorem coeSubmodule_span (s : Set R) :
   rfl
 #align is_localization.coe_submodule_span IsLocalization.coeSubmodule_span
 
+/- warning: is_localization.coe_submodule_span_singleton -> IsLocalization.coeSubmodule_span_singleton is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.hasSingleton.{u1} R) x))) (Submodule.span.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (Singleton.singleton.{u2, u2} S (Set.{u2} S) (Set.hasSingleton.{u2} S) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) x)))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] (x : R), Eq.{succ u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Singleton.singleton.{u1, u1} R (Set.{u1} R) (Set.instSingletonSet.{u1} R) x))) (Submodule.span.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (NonAssocRing.toNonUnitalNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Ring.toNonAssocRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (CommRing.toCommSemiring.{u1} R _inst_1) _inst_2 _inst_3) (Singleton.singleton.{u2, u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x) (Set.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (Set.instSingletonSet.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) x)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => S) _x) (MulHomClass.toFunLike.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u1 u2, u1, u2} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (RingHom.instRingHomClassRingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) x)))
+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_span_singleton IsLocalization.coeSubmodule_span_singletonₓ'. -/
 @[simp]
 theorem coeSubmodule_span_singleton (x : R) :
     coeSubmodule S (Ideal.span {x}) = Submodule.span R {(algebraMap R S) x} := by
@@ -101,6 +157,12 @@ section
 
 include M
 
+/- warning: is_localization.is_noetherian_ring -> IsLocalization.isNoetherianRing is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : 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)))))) (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (IsNoetherianRing.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) -> (IsNoetherianRing.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (IsNoetherianRing.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) -> (IsNoetherianRing.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))
+Case conversion may be inaccurate. Consider using '#align is_localization.is_noetherian_ring IsLocalization.isNoetherianRingₓ'. -/
 theorem isNoetherianRing (h : IsNoetherianRing R) : IsNoetherianRing S :=
   by
   rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at h⊢
@@ -111,12 +173,24 @@ end
 
 variable {S Q M}
 
+/- warning: is_localization.coe_submodule_le_coe_submodule -> IsLocalization.coeSubmodule_le_coeSubmodule is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (forall {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (forall {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 J)) (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J))
+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmoduleₓ'. -/
 @[mono]
 theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal R} :
     coeSubmodule S I ≤ coeSubmodule S J ↔ I ≤ J :=
   Submodule.map_le_map_iff_of_injective (IsLocalization.injective _ h) _ _
 #align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmodule
 
+/- warning: is_localization.coe_submodule_strict_mono -> IsLocalization.coeSubmodule_strictMono is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (StrictMono.{u1, u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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))))))) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (Submodule.completeLattice.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (StrictMono.{u2, u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (Submodule.completeLattice.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_strict_mono IsLocalization.coeSubmodule_strictMonoₓ'. -/
 @[mono]
 theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
     StrictMono (coeSubmodule S : Ideal R → Submodule R S) :=
@@ -125,11 +199,23 @@ theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
 
 variable (S) {Q M}
 
+/- warning: is_localization.coe_submodule_injective -> IsLocalization.coeSubmodule_injective is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3], (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Function.Injective.{succ u1, succ u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R S (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} S (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3], (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R S (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} S (NonAssocRing.toNonUnitalNonAssocRing.{u1} S (Ring.toNonAssocRing.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3))
+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_injective IsLocalization.coeSubmodule_injectiveₓ'. -/
 theorem coeSubmodule_injective (h : M ≤ nonZeroDivisors R) :
     Function.Injective (coeSubmodule S : Ideal R → Submodule R S) :=
   injective_of_le_imp_le _ fun _ _ => (coeSubmodule_le_coeSubmodule h).mp
 #align is_localization.coe_submodule_injective IsLocalization.coeSubmodule_injective
 
+/- warning: is_localization.coe_submodule_is_principal -> IsLocalization.coeSubmodule_isPrincipal is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : 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)))))} (S : Type.{u2}) [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (LE.le.{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)))))) (Preorder.toLE.{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)))))) (PartialOrder.toPreorder.{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)))))) (CompleteSemilatticeInf.toPartialOrder.{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)))))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))) (Submonoid.completeLattice.{u1} R (MulZeroOneClass.toMulOneClass.{u1} R (NonAssocSemiring.toMulZeroOneClass.{u1} R (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))))) M (nonZeroDivisors.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u1, u2} R S (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))} (S : Type.{u1}) [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, (LE.le.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Preorder.toLE.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (PartialOrder.toPreorder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (Submonoid.instCompleteLatticeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))))) M (nonZeroDivisors.{u2} R (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) -> (Iff (Submodule.IsPrincipal.{u2, u1} R S (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} S (CommRing.toRing.{u1} S _inst_2)) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) _inst_2 _inst_3) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 S _inst_2 _inst_3 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I))
+Case conversion may be inaccurate. Consider using '#align is_localization.coe_submodule_is_principal IsLocalization.coeSubmodule_isPrincipalₓ'. -/
 theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
     (coeSubmodule S I).IsPrincipal ↔ I.IsPrincipal :=
   by
@@ -144,6 +230,12 @@ theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
 
 variable {S} (M)
 
+/- warning: is_localization.mem_span_iff -> IsLocalization.mem_span_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : 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)))))) {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {N : Type.{u3}} [_inst_9 : AddCommGroup.{u3} N] [_inst_10 : Module.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_11 : Module.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_12 : IsScalarTower.{u1, u2, u3} R S N (SMulZeroClass.toHasSmul.{u1, u2} R S (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R S (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3))))) (SMulZeroClass.toHasSmul.{u2, u3} S N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (SMulWithZero.toSmulZeroClass.{u2, u3} S N (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (MulActionWithZero.toSMulWithZero.{u2, u3} S N (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (Module.toMulActionWithZero.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)))) (SMulZeroClass.toHasSmul.{u1, u3} R N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (SMulWithZero.toSmulZeroClass.{u1, u3} R N (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (MulActionWithZero.toSMulWithZero.{u1, u3} R N (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (Module.toMulActionWithZero.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10))))] {x : N} {a : Set.{u3} N}, Iff (Membership.Mem.{u3, u3} N (Submodule.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) (SetLike.hasMem.{u3, u3} (Submodule.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) N (Submodule.setLike.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)) x (Submodule.span.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11 a)) (Exists.{succ u3} N (fun (y : N) => Exists.{0} (Membership.Mem.{u3, u3} N (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) N (Submodule.setLike.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10)) y (Submodule.span.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10 a)) (fun (H : Membership.Mem.{u3, u3} N (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) N (Submodule.setLike.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10)) y (Submodule.span.{u1, u3} R N (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10 a)) => Exists.{succ u1} (coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) (fun (z : coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) => Eq.{succ u3} N x (SMul.smul.{u2, u3} S N (SMulZeroClass.toHasSmul.{u2, u3} S N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (SMulWithZero.toSmulZeroClass.{u2, u3} S N (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (MulActionWithZero.toSMulWithZero.{u2, u3} S N (Semiring.toMonoidWithZero.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)))) (Module.toMulActionWithZero.{u2, u3} S N (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)))) (IsLocalization.mk'.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3 _inst_8 (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) z) y)))))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {N : Type.{u3}} [_inst_9 : AddCommGroup.{u3} N] [_inst_10 : Module.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_11 : Module.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9)] [_inst_12 : IsScalarTower.{u2, u1, u3} R S N (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3) (SMulZeroClass.toSMul.{u1, u3} S N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S N (CommMonoidWithZero.toZero.{u1} S (CommSemiring.toCommMonoidWithZero.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S N (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)))) (SMulZeroClass.toSMul.{u2, u3} R N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R N (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R N (Semiring.toMonoidWithZero.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10))))] {x : N} {a : Set.{u3} N}, Iff (Membership.mem.{u3, u3} N (Submodule.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) (SetLike.instMembership.{u3, u3} (Submodule.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11) N (Submodule.setLike.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11)) x (Submodule.span.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11 a)) (Exists.{succ u3} N (fun (y : N) => And (Membership.mem.{u3, u3} N (Submodule.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10) N (Submodule.setLike.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10)) y (Submodule.span.{u2, u3} R N (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_10 a)) (Exists.{succ u2} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) x M)) (fun (z : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) x M)) => Eq.{succ u3} N x (HSMul.hSMul.{u1, u3, u3} S N N (instHSMul.{u1, u3} S N (SMulZeroClass.toSMul.{u1, u3} S N (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (SMulWithZero.toSMulZeroClass.{u1, u3} S N (CommMonoidWithZero.toZero.{u1} S (CommSemiring.toCommMonoidWithZero.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (MulActionWithZero.toSMulWithZero.{u1, u3} S N (Semiring.toMonoidWithZero.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (NegZeroClass.toZero.{u3} N (SubNegZeroMonoid.toNegZeroClass.{u3} N (SubtractionMonoid.toSubNegZeroMonoid.{u3} N (SubtractionCommMonoid.toSubtractionMonoid.{u3} N (AddCommGroup.toDivisionAddCommMonoid.{u3} N _inst_9))))) (Module.toMulActionWithZero.{u1, u3} S N (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} N _inst_9) _inst_11))))) (IsLocalization.mk'.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3 _inst_8 (OfNat.ofNat.{u2} R 1 (One.toOfNat1.{u2} R (NonAssocRing.toOne.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))) z) y)))))
+Case conversion may be inaccurate. Consider using '#align is_localization.mem_span_iff IsLocalization.mem_span_iffₓ'. -/
 theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]
     {x : N} {a : Set N} :
     x ∈ Submodule.span S a ↔ ∃ y ∈ Submodule.span R a, ∃ z : M, x = mk' S 1 z • y :=
@@ -173,6 +265,12 @@ theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [Is
     exact Submodule.smul_mem _ _ (Submodule.span_subset_span R S _ hy)
 #align is_localization.mem_span_iff IsLocalization.mem_span_iff
 
+/- warning: is_localization.mem_span_map -> IsLocalization.mem_span_map is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (M : 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)))))) {S : Type.{u2}} [_inst_2 : CommRing.{u2} S] [_inst_3 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_8 : IsLocalization.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3] {x : S} {a : Set.{u1} R}, Iff (Membership.Mem.{u2, u2} S (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) (SetLike.hasMem.{u2, u2} (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))) S (Submodule.setLike.{u2, u2} S S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) (Semiring.toModule.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))))) x (Ideal.span.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) (Set.image.{u1, u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_3)) a))) (Exists.{succ u1} R (fun (y : R) => Exists.{0} (Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{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))))) y (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) a)) (fun (H : Membership.Mem.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.hasMem.{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))))) y (Ideal.span.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) a)) => Exists.{succ u1} (coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) (fun (z : coeSort.{succ u1, succ (succ 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)))))) Type.{u1} (SetLike.hasCoeToSort.{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))))))) M) => Eq.{succ u2} S x (IsLocalization.mk'.{u1, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) M S (CommRing.toCommSemiring.{u2} S _inst_2) _inst_3 _inst_8 y z)))))
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] (M : Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) {S : Type.{u1}} [_inst_2 : CommRing.{u1} S] [_inst_3 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))] [_inst_8 : IsLocalization.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3] {x : S} {a : Set.{u2} R}, Iff (Membership.mem.{u1, u1} S (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (SetLike.instMembership.{u1, u1} (Ideal.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) S (Submodule.setLike.{u1, u1} S S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (Semiring.toModule.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) x (Ideal.span.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) (Set.image.{u2, u1} R S (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) 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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) (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_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)))) R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2))))))) (algebraMap.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} S (CommRing.toRing.{u1} S _inst_2)) _inst_3)) a))) (Exists.{succ u2} R (fun (y : R) => And (Membership.mem.{u2, u2} R (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (SetLike.instMembership.{u2, u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) R (Submodule.setLike.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) y (Ideal.span.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) a)) (Exists.{succ u2} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) x M)) (fun (z : Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) (SetLike.instMembership.{u2, u2} (Submonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1)))))) R (Submonoid.instSetLikeSubmonoid.{u2} R (MulZeroOneClass.toMulOneClass.{u2} R (NonAssocSemiring.toMulZeroOneClass.{u2} R (NonAssocRing.toNonAssocSemiring.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) x M)) => Eq.{succ u1} S x (IsLocalization.mk'.{u2, u1} R (CommRing.toCommSemiring.{u2} R _inst_1) M S (CommRing.toCommSemiring.{u1} S _inst_2) _inst_3 _inst_8 y z)))))
+Case conversion may be inaccurate. Consider using '#align is_localization.mem_span_map IsLocalization.mem_span_mapₓ'. -/
 theorem mem_span_map {x : S} {a : Set R} :
     x ∈ Ideal.span (algebraMap R S '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z : M, x = mk' S y z :=
   by
@@ -199,12 +297,24 @@ section CommRing
 
 variable [CommRing K] [Algebra R K] [IsFractionRing R K] [Algebra A K] [IsFractionRing A K]
 
+/- warning: is_fraction_ring.coe_submodule_le_coe_submodule -> IsFractionRing.coeSubmodule_le_coeSubmodule is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {K : Type.{u2}} [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))} {J : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (LE.le.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 J)) (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))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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)))))))) I J)
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))} {J : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (LE.le.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (Submodule.completeLattice.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 J)) (LE.le.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Preorder.toLE.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)))))))) I J)
+Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_le_coe_submodule IsFractionRing.coeSubmodule_le_coeSubmoduleₓ'. -/
 @[simp, mono]
 theorem coeSubmodule_le_coeSubmodule {I J : Ideal R} :
     coeSubmodule K I ≤ coeSubmodule K J ↔ I ≤ J :=
   IsLocalization.coeSubmodule_le_coeSubmodule le_rfl
 #align is_fraction_ring.coe_submodule_le_coe_submodule IsFractionRing.coeSubmodule_le_coeSubmodule
 
+/- warning: is_fraction_ring.coe_submodule_strict_mono -> IsFractionRing.coeSubmodule_strictMono is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {K : Type.{u2}} [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7], StrictMono.{u1, u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteSemilatticeInf.toPartialOrder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (CompleteLattice.toCompleteSemilatticeInf.{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))))))) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (Submodule.completeLattice.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7))))) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7)
+but is expected to have type
+  forall {R : Type.{u2}} [_inst_1 : CommRing.{u2} R] {K : Type.{u1}} [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], StrictMono.{u2, u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (PartialOrder.toPreorder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.completeLattice.{u2, u2} R R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))))))) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (Submodule.completeLattice.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7))))) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
+Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_strict_mono IsFractionRing.coeSubmodule_strictMonoₓ'. -/
 @[mono]
 theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submodule R K) :=
   strictMono_of_le_iff_le fun _ _ => coeSubmodule_le_coeSubmodule.symm
@@ -212,10 +322,22 @@ theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submo
 
 variable (R K)
 
+/- warning: is_fraction_ring.coe_submodule_injective -> IsFractionRing.coeSubmodule_injective is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] (K : Type.{u2}) [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7], Function.Injective.{succ u1, succ u2} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Submodule.{u1, u2} R K (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} K (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6))))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7)) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7)
+but is expected to have type
+  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7], Function.Injective.{succ u2, succ u1} (Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) (Submodule.{u2, u1} R K (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (CommRing.toRing.{u1} K _inst_6))))) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7)) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7)
+Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_injective IsFractionRing.coeSubmodule_injectiveₓ'. -/
 theorem coeSubmodule_injective : Function.Injective (coeSubmodule K : Ideal R → Submodule R K) :=
   injective_of_le_imp_le _ fun _ _ => coeSubmodule_le_coeSubmodule.mp
 #align is_fraction_ring.coe_submodule_injective IsFractionRing.coeSubmodule_injective
 
+/- warning: is_fraction_ring.coe_submodule_is_principal -> IsFractionRing.coeSubmodule_isPrincipal is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) [_inst_1 : CommRing.{u1} R] (K : Type.{u2}) [_inst_6 : CommRing.{u2} K] [_inst_7 : Algebra.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6))] [_inst_8 : IsFractionRing.{u1, u2} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, Iff (Submodule.IsPrincipal.{u1, u2} R K (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u2} K (NonAssocRing.toNonUnitalNonAssocRing.{u2} K (Ring.toNonAssocRing.{u2} K (CommRing.toRing.{u2} K _inst_6)))) (Algebra.toModule.{u1, u2} R K (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} K (CommRing.toRing.{u2} K _inst_6)) _inst_7) (IsLocalization.coeSubmodule.{u1, u2} R _inst_1 K _inst_6 _inst_7 I)) (Submodule.IsPrincipal.{u1, u1} R R (CommRing.toRing.{u1} R _inst_1) (NonUnitalNonAssocRing.toAddCommGroup.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) I)
+but is expected to have type
+  forall (R : Type.{u2}) [_inst_1 : CommRing.{u2} R] (K : Type.{u1}) [_inst_6 : CommRing.{u1} K] [_inst_7 : Algebra.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} K (CommRing.toRing.{u1} K _inst_6))] [_inst_8 : IsFractionRing.{u2, u1} R _inst_1 K _inst_6 _inst_7] {I : Ideal.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))}, Iff (Submodule.IsPrincipal.{u2, u1} R K (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u1} K (CommRing.toRing.{u1} K _inst_6)) (_private.Mathlib.RingTheory.Ideal.Operations.0.Ideal.instModuleToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonUnitalRingToNonUnitalCommRing.{u2, u1} R K (CommRing.toCommSemiring.{u2} R _inst_1) _inst_6 _inst_7) (IsLocalization.coeSubmodule.{u2, u1} R _inst_1 K _inst_6 _inst_7 I)) (Submodule.IsPrincipal.{u2, u2} R R (CommRing.toRing.{u2} R _inst_1) (Ring.toAddCommGroup.{u2} R (CommRing.toRing.{u2} R _inst_1)) (Semiring.toModule.{u2} R (Ring.toSemiring.{u2} R (CommRing.toRing.{u2} R _inst_1))) I)
+Case conversion may be inaccurate. Consider using '#align is_fraction_ring.coe_submodule_is_principal IsFractionRing.coeSubmodule_isPrincipalₓ'. -/
 @[simp]
 theorem coeSubmodule_isPrincipal {I : Ideal R} : (coeSubmodule K I).IsPrincipal ↔ I.IsPrincipal :=
   IsLocalization.coeSubmodule_isPrincipal _ le_rfl

1ebb2060

bump to nightly-2023-02-21-16

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

1107b979

Changes in mathlib4

mathlib3

mathlib4

1ebb2060

chore: adapt to multiple goal linter 3 (#12372)

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

d29b3780

Diff

@@ -138,8 +138,9 @@ variable {S} (M)
 theorem mem_span_iff {N : Type*} [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]
     {x : N} {a : Set N} :
     x ∈ Submodule.span S a ↔ ∃ y ∈ Submodule.span R a, ∃ z : M, x = mk' S 1 z • y := by
-  constructor; intro h
-  · refine' Submodule.span_induction h _ _ _ _
+  constructor
+  · intro h
+    refine' Submodule.span_induction h _ _ _ _
     · rintro x hx
       exact ⟨x, Submodule.subset_span hx, 1, by rw [mk'_one, _root_.map_one, one_smul]⟩
     · exact ⟨0, Submodule.zero_mem _, 1, by rw [mk'_one, _root_.map_one, one_smul]⟩

1ebb2060

chore(*): remove empty lines between variable statements (#11418)

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)

75c86f04

Diff

@@ -23,7 +23,6 @@ commutative ring, field of fractions
 
 
 variable {R : Type*} [CommRing R] (M : Submonoid R) (S : Type*) [CommRing S]
-
 variable [Algebra R S] {P : Type*} [CommRing P]
 
 namespace IsLocalization
@@ -86,11 +85,8 @@ theorem coeSubmodule_span_singleton (x : R) :
 #align is_localization.coe_submodule_span_singleton IsLocalization.coeSubmodule_span_singleton
 
 variable {g : R →+* P}
-
 variable {T : Submonoid P} (hy : M ≤ T.comap g) {Q : Type*} [CommRing Q]
-
 variable [Algebra P Q] [IsLocalization T Q]
-
 variable [IsLocalization M S]
 
 section

1ebb2060

feat: sum and product of commuting semisimple endomorphisms (#10808)

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>

0b52a6a5

Diff

@@ -5,7 +5,7 @@ Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baan
 -/
 import Mathlib.RingTheory.Localization.FractionRing
 import Mathlib.RingTheory.Localization.Ideal
-import Mathlib.RingTheory.PrincipalIdealDomain
+import Mathlib.RingTheory.Noetherian
 
 #align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"1ebb20602a8caef435ce47f6373e1aa40851a177"

1ebb2060

chore: classify simp can do this porting notes (#10619)

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

de765f60

Diff

@@ -79,7 +79,7 @@ theorem coeSubmodule_span (s : Set R) :
   rfl
 #align is_localization.coe_submodule_span IsLocalization.coeSubmodule_span
 
--- @[simp] -- Porting note: simp can prove this
+-- @[simp] -- Porting note (#10618): simp can prove this
 theorem coeSubmodule_span_singleton (x : R) :
     coeSubmodule S (Ideal.span {x}) = Submodule.span R {(algebraMap R S) x} := by
   rw [coeSubmodule_span, Set.image_singleton]

1ebb2060

refactor(Data/FunLike): use unbundled inheritance from FunLike (#8386)

The FunLike hierarchy is very big and gets scanned through each time we need a coercion (via the CoeFun instance). It looks like unbundled inheritance suits Lean 4 better here. The only class that still extends FunLike is EquivLike, since that has a custom coe_injective' field that is easier to implement. All other classes should take FunLike or EquivLike as a parameter.

Zulip thread

Important changes

Previously, morphism classes would be Type-valued and extend FunLike:

/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
  extends FunLike F A B :=
(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))

After this PR, they should be Prop-valued and take FunLike as a parameter:

/-- `MyHomClass F A B` states that `F` is a type of `MyClass.op`-preserving morphisms.
You should extend this class when you extend `MyHom`. -/
class MyHomClass (F : Type*) (A B : outParam <| Type*) [MyClass A] [MyClass B]
  [FunLike F A B] : Prop :=
(map_op : ∀ (f : F) (x y : A), f (MyClass.op x y) = MyClass.op (f x) (f y))

(Note that A B stay marked as outParam even though they are not purely required to be so due to the FunLike parameter already filling them in. This is required to see through type synonyms, which is important in the category theory library. Also, I think keeping them as outParam is slightly faster.)

Similarly, MyEquivClass should take EquivLike as a parameter.

As a result, every mention of [MyHomClass F A B] should become [FunLike F A B] [MyHomClass F A B].

Remaining issues

Slower (failing) search

While overall this gives some great speedups, there are some cases that are noticeably slower. In particular, a failing application of a lemma such as map_mul is more expensive. This is due to suboptimal processing of arguments. For example:

variable [FunLike F M N] [Mul M] [Mul N] (f : F) (x : M) (y : M)

theorem map_mul [MulHomClass F M N] : f (x * y) = f x * f y

example [AddHomClass F A B] : f (x * y) = f x * f y := map_mul f _ _

Before this PR, applying map_mul f gives the goals [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]. Since M and N are out_params, [MulHomClass F ?M ?N] is synthesized first, supplies values for ?M and ?N and then the Mul M and Mul N instances can be found.

After this PR, the goals become [FunLike F ?M ?N] [Mul ?M] [Mul ?N] [MulHomClass F ?M ?N]. Now [FunLike F ?M ?N] is synthesized first, supplies values for ?M and ?N and then the Mul M and Mul N instances can be found, before trying MulHomClass F M N which fails. Since the Mul hierarchy is very big, this can be slow to fail, especially when there is no such Mul instance.

A long-term but harder to achieve solution would be to specify the order in which instance goals get solved. For example, we'd like to change the arguments to map_mul to look like [FunLike F M N] [Mul M] [Mul N] [highPriority <| MulHomClass F M N] because MulHomClass fails or succeeds much faster than the others.

As a consequence, the simpNF linter is much slower since by design it tries and fails to apply many map_ lemmas. The same issue occurs a few times in existing calls to simp [map_mul], where map_mul is tried "too soon" and fails. Thanks to the speedup of leanprover/lean4#2478 the impact is very limited, only in files that already were close to the timeout.

`simp` not firing sometimes

This affects map_smulₛₗ and related definitions. For simp lemmas Lean apparently uses a slightly different mechanism to find instances, so that rw can find every argument to map_smulₛₗ successfully but simp can't: leanprover/lean4#3701.

Missing instances due to unification failing

Especially in the category theory library, we might sometimes have a type A which is also accessible as a synonym (Bundled A hA).1. Instance synthesis doesn't always work if we have f : A →* B but x * y : (Bundled A hA).1 or vice versa. This seems to be mostly fixed by keeping A B as outParams in MulHomClass F A B. (Presumably because Lean will do a definitional check A =?= (Bundled A hA).1 instead of using the syntax in the discrimination tree.)

Workaround for issues

The timeouts can be worked around for now by specifying which map_mul we mean, either as map_mul f for some explicit f, or as e.g. MonoidHomClass.map_mul.

map_smulₛₗ not firing as simp lemma can be worked around by going back to the pre-FunLike situation and making LinearMap.map_smulₛₗ a simp lemma instead of the generic map_smulₛₗ. Writing simp [map_smulₛₗ _] also works.

Co-authored-by: Matthew Ballard <matt@mrb.email> Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott@tqft.net> Co-authored-by: Anne Baanen <Vierkantor@users.noreply.github.com>

c6a73d4f

Diff

@@ -107,7 +107,9 @@ variable {S M}
 @[mono]
 theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal R} :
     coeSubmodule S I ≤ coeSubmodule S J ↔ I ≤ J :=
-  Submodule.map_le_map_iff_of_injective (IsLocalization.injective _ h) _ _
+  -- Note: #8386 had to specify the value of `f` here:
+  Submodule.map_le_map_iff_of_injective (f := Algebra.linearMap R S) (IsLocalization.injective _ h)
+    _ _
 #align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmodule

1ebb2060

chore: banish Type _ and Sort _ (#6499)

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

This has nice performance benefits.

32de3f67

Diff

@@ -22,9 +22,9 @@ commutative ring, field of fractions
 -/
 
 
-variable {R : Type _} [CommRing R] (M : Submonoid R) (S : Type _) [CommRing S]
+variable {R : Type*} [CommRing R] (M : Submonoid R) (S : Type*) [CommRing S]
 
-variable [Algebra R S] {P : Type _} [CommRing P]
+variable [Algebra R S] {P : Type*} [CommRing P]
 
 namespace IsLocalization
 
@@ -87,7 +87,7 @@ theorem coeSubmodule_span_singleton (x : R) :
 
 variable {g : R →+* P}
 
-variable {T : Submonoid P} (hy : M ≤ T.comap g) {Q : Type _} [CommRing Q]
+variable {T : Submonoid P} (hy : M ≤ T.comap g) {Q : Type*} [CommRing Q]
 
 variable [Algebra P Q] [IsLocalization T Q]
 
@@ -137,7 +137,7 @@ theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
 
 variable {S} (M)
 
-theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]
+theorem mem_span_iff {N : Type*} [AddCommGroup N] [Module R N] [Module S N] [IsScalarTower R S N]
     {x : N} {a : Set N} :
     x ∈ Submodule.span S a ↔ ∃ y ∈ Submodule.span R a, ∃ z : M, x = mk' S 1 z • y := by
   constructor; intro h
@@ -182,7 +182,7 @@ namespace IsFractionRing
 
 open IsLocalization
 
-variable {A K : Type _} [CommRing A]
+variable {A K : Type*} [CommRing A]
 
 section CommRing

1ebb2060

chore: script to replace headers with #align_import statements (#5979)

depends on: #5966

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

89aa3a3b

Diff

@@ -2,16 +2,13 @@
 Copyright (c) 2018 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-
-! This file was ported from Lean 3 source module ring_theory.localization.submodule
-! leanprover-community/mathlib commit 1ebb20602a8caef435ce47f6373e1aa40851a177
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.RingTheory.Localization.FractionRing
 import Mathlib.RingTheory.Localization.Ideal
 import Mathlib.RingTheory.PrincipalIdealDomain
 
+#align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"1ebb20602a8caef435ce47f6373e1aa40851a177"
+
 /-!
 # Submodules in localizations of commutative rings

1ebb2060

chore: clean up spacing around at and goals (#5387)

Changes are of the form

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

2a870323

Diff

@@ -99,7 +99,7 @@ variable [IsLocalization M S]
 section
 
 theorem isNoetherianRing (h : IsNoetherianRing R) : IsNoetherianRing S := by
-  rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at h⊢
+  rw [isNoetherianRing_iff, isNoetherian_iff_wellFounded] at h ⊢
   exact OrderEmbedding.wellFounded (IsLocalization.orderEmbedding M S).dual h
 #align is_localization.is_noetherian_ring IsLocalization.isNoetherianRing

1ebb2060

chore: reenable eta, bump to nightly 2023-05-16 (#3414)

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>

a6e1045f

Diff

@@ -31,7 +31,6 @@ variable [Algebra R S] {P : Type _} [CommRing P]
 
 namespace IsLocalization
 
-set_option synthInstance.etaExperiment true in -- Porting note: gets around lean4#2074
 -- This was previously a `hasCoe` instance, but if `S = R` then this will loop.
 -- It could be a `hasCoeT` instance, but we keep it explicit here to avoid slowing down
 -- the rest of the library.
@@ -40,50 +39,42 @@ def coeSubmodule (I : Ideal R) : Submodule R S :=
   Submodule.map (Algebra.linearMap R S) I
 #align is_localization.coe_submodule IsLocalization.coeSubmodule
 
-set_option synthInstance.etaExperiment true in
 theorem mem_coeSubmodule (I : Ideal R) {x : S} :
     x ∈ coeSubmodule S I ↔ ∃ y : R, y ∈ I ∧ algebraMap R S y = x :=
   Iff.rfl
 #align is_localization.mem_coe_submodule IsLocalization.mem_coeSubmodule
 
-set_option synthInstance.etaExperiment true in -- Porting note: gets around lean4#2074
 theorem coeSubmodule_mono {I J : Ideal R} (h : I ≤ J) : coeSubmodule S I ≤ coeSubmodule S J :=
   Submodule.map_mono h
 #align is_localization.coe_submodule_mono IsLocalization.coeSubmodule_mono
 
-set_option synthInstance.etaExperiment true in -- Porting note: gets around lean4#2074
 @[simp]
 theorem coeSubmodule_bot : coeSubmodule S (⊥ : Ideal R) = ⊥ := by
   rw [coeSubmodule, Submodule.map_bot]
 #align is_localization.coe_submodule_bot IsLocalization.coeSubmodule_bot
 
-set_option synthInstance.etaExperiment true in -- Porting note: gets around lean4#2074
 @[simp]
 theorem coeSubmodule_top : coeSubmodule S (⊤ : Ideal R) = 1 := by
   rw [coeSubmodule, Submodule.map_top, Submodule.one_eq_range]
 #align is_localization.coe_submodule_top IsLocalization.coeSubmodule_top
 
-set_option synthInstance.etaExperiment true in -- Porting note: gets around lean4#2074
 @[simp]
 theorem coeSubmodule_sup (I J : Ideal R) :
     coeSubmodule S (I ⊔ J) = coeSubmodule S I ⊔ coeSubmodule S J :=
   Submodule.map_sup _ _ _
 #align is_localization.coe_submodule_sup IsLocalization.coeSubmodule_sup
 
-set_option synthInstance.etaExperiment true in -- Porting note: gets around lean4#2074
 @[simp]
 theorem coeSubmodule_mul (I J : Ideal R) :
     coeSubmodule S (I * J) = coeSubmodule S I * coeSubmodule S J :=
   Submodule.map_mul _ _ (Algebra.ofId R S)
 #align is_localization.coe_submodule_mul IsLocalization.coeSubmodule_mul
 
-set_option synthInstance.etaExperiment true in
 theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R) :
     Submodule.FG (coeSubmodule S I) ↔ Submodule.FG I :=
   ⟨Submodule.fg_of_fg_map _ (LinearMap.ker_eq_bot.mpr hS), Submodule.FG.map _⟩
 #align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fg
 
-set_option synthInstance.etaExperiment true in
 @[simp]
 theorem coeSubmodule_span (s : Set R) :
     coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s) := by
@@ -91,7 +82,6 @@ theorem coeSubmodule_span (s : Set R) :
   rfl
 #align is_localization.coe_submodule_span IsLocalization.coeSubmodule_span
 
-set_option synthInstance.etaExperiment true in
 -- @[simp] -- Porting note: simp can prove this
 theorem coeSubmodule_span_singleton (x : R) :
     coeSubmodule S (Ideal.span {x}) = Submodule.span R {(algebraMap R S) x} := by
@@ -117,14 +107,12 @@ end
 
 variable {S M}
 
-set_option synthInstance.etaExperiment true in -- Porting note: gets around lean4#2074
 @[mono]
 theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal R} :
     coeSubmodule S I ≤ coeSubmodule S J ↔ I ≤ J :=
   Submodule.map_le_map_iff_of_injective (IsLocalization.injective _ h) _ _
 #align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmodule
 
-set_option synthInstance.etaExperiment true in
 
 @[mono]
 theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
@@ -134,13 +122,11 @@ theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
 
 variable (S)
 
-set_option synthInstance.etaExperiment true in
 theorem coeSubmodule_injective (h : M ≤ nonZeroDivisors R) :
     Function.Injective (coeSubmodule S : Ideal R → Submodule R S) :=
   injective_of_le_imp_le _ fun hl => (coeSubmodule_le_coeSubmodule h).mp hl
 #align is_localization.coe_submodule_injective IsLocalization.coeSubmodule_injective
 
-set_option synthInstance.etaExperiment true in
 theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
     (coeSubmodule S I).IsPrincipal ↔ I.IsPrincipal := by
   constructor <;> rintro ⟨⟨x, hx⟩⟩
@@ -180,7 +166,6 @@ theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [Is
     exact Submodule.smul_mem _ _ (Submodule.span_subset_span R S _ hy)
 #align is_localization.mem_span_iff IsLocalization.mem_span_iff
 
-set_option synthInstance.etaExperiment true in
 theorem mem_span_map {x : S} {a : Set R} :
     x ∈ Ideal.span (algebraMap R S '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z : M, x = mk' S y z := by
   refine' (mem_span_iff M).trans _
@@ -206,14 +191,12 @@ section CommRing
 
 variable [CommRing K] [Algebra R K] [IsFractionRing R K] [Algebra A K] [IsFractionRing A K]
 
-set_option synthInstance.etaExperiment true in
 @[simp, mono]
 theorem coeSubmodule_le_coeSubmodule {I J : Ideal R} :
     coeSubmodule K I ≤ coeSubmodule K J ↔ I ≤ J :=
   IsLocalization.coeSubmodule_le_coeSubmodule le_rfl
 #align is_fraction_ring.coe_submodule_le_coe_submodule IsFractionRing.coeSubmodule_le_coeSubmodule
 
-set_option synthInstance.etaExperiment true in
 @[mono]
 theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submodule R K) :=
   strictMono_of_le_iff_le fun _ _ => coeSubmodule_le_coeSubmodule.symm
@@ -221,12 +204,10 @@ theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submo
 
 variable (R K)
 
-set_option synthInstance.etaExperiment true in
 theorem coeSubmodule_injective : Function.Injective (coeSubmodule K : Ideal R → Submodule R K) :=
   injective_of_le_imp_le _ fun hl => coeSubmodule_le_coeSubmodule.mp hl
 #align is_fraction_ring.coe_submodule_injective IsFractionRing.coeSubmodule_injective
 
-set_option synthInstance.etaExperiment true in
 @[simp]
 theorem coeSubmodule_isPrincipal {I : Ideal R} : (coeSubmodule K I).IsPrincipal ↔ I.IsPrincipal :=
   IsLocalization.coeSubmodule_isPrincipal _ le_rfl

1ebb2060

refactor: rename Fg to FG (#3948)

Please refer to this Zulip thread: https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Naming.20convention/near/357712556

000e226b

Diff

@@ -79,8 +79,8 @@ theorem coeSubmodule_mul (I J : Ideal R) :
 
 set_option synthInstance.etaExperiment true in
 theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R) :
-    Submodule.Fg (coeSubmodule S I) ↔ Submodule.Fg I :=
-  ⟨Submodule.fg_of_fg_map _ (LinearMap.ker_eq_bot.mpr hS), Submodule.Fg.map _⟩
+    Submodule.FG (coeSubmodule S I) ↔ Submodule.FG I :=
+  ⟨Submodule.fg_of_fg_map _ (LinearMap.ker_eq_bot.mpr hS), Submodule.FG.map _⟩
 #align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fg
 
 set_option synthInstance.etaExperiment true in

1ebb2060

chore: use etaExperiment rather than hacking with instances (#3668)

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>

0f225a7f

Diff

@@ -40,6 +40,7 @@ def coeSubmodule (I : Ideal R) : Submodule R S :=
   Submodule.map (Algebra.linearMap R S) I
 #align is_localization.coe_submodule IsLocalization.coeSubmodule
 
+set_option synthInstance.etaExperiment true in
 theorem mem_coeSubmodule (I : Ideal R) {x : S} :
     x ∈ coeSubmodule S I ↔ ∃ y : R, y ∈ I ∧ algebraMap R S y = x :=
   Iff.rfl
@@ -76,11 +77,13 @@ theorem coeSubmodule_mul (I J : Ideal R) :
   Submodule.map_mul _ _ (Algebra.ofId R S)
 #align is_localization.coe_submodule_mul IsLocalization.coeSubmodule_mul
 
+set_option synthInstance.etaExperiment true in
 theorem coeSubmodule_fg (hS : Function.Injective (algebraMap R S)) (I : Ideal R) :
     Submodule.Fg (coeSubmodule S I) ↔ Submodule.Fg I :=
   ⟨Submodule.fg_of_fg_map _ (LinearMap.ker_eq_bot.mpr hS), Submodule.Fg.map _⟩
 #align is_localization.coe_submodule_fg IsLocalization.coeSubmodule_fg
 
+set_option synthInstance.etaExperiment true in
 @[simp]
 theorem coeSubmodule_span (s : Set R) :
     coeSubmodule S (Ideal.span s) = Submodule.span R (algebraMap R S '' s) := by
@@ -88,6 +91,7 @@ theorem coeSubmodule_span (s : Set R) :
   rfl
 #align is_localization.coe_submodule_span IsLocalization.coeSubmodule_span
 
+set_option synthInstance.etaExperiment true in
 -- @[simp] -- Porting note: simp can prove this
 theorem coeSubmodule_span_singleton (x : R) :
     coeSubmodule S (Ideal.span {x}) = Submodule.span R {(algebraMap R S) x} := by
@@ -120,6 +124,8 @@ theorem coeSubmodule_le_coeSubmodule (h : M ≤ nonZeroDivisors R) {I J : Ideal
   Submodule.map_le_map_iff_of_injective (IsLocalization.injective _ h) _ _
 #align is_localization.coe_submodule_le_coe_submodule IsLocalization.coeSubmodule_le_coeSubmodule
 
+set_option synthInstance.etaExperiment true in
+
 @[mono]
 theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
     StrictMono (coeSubmodule S : Ideal R → Submodule R S) :=
@@ -128,11 +134,13 @@ theorem coeSubmodule_strictMono (h : M ≤ nonZeroDivisors R) :
 
 variable (S)
 
+set_option synthInstance.etaExperiment true in
 theorem coeSubmodule_injective (h : M ≤ nonZeroDivisors R) :
     Function.Injective (coeSubmodule S : Ideal R → Submodule R S) :=
   injective_of_le_imp_le _ fun hl => (coeSubmodule_le_coeSubmodule h).mp hl
 #align is_localization.coe_submodule_injective IsLocalization.coeSubmodule_injective
 
+set_option synthInstance.etaExperiment true in
 theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
     (coeSubmodule S I).IsPrincipal ↔ I.IsPrincipal := by
   constructor <;> rintro ⟨⟨x, hx⟩⟩
@@ -172,6 +180,7 @@ theorem mem_span_iff {N : Type _} [AddCommGroup N] [Module R N] [Module S N] [Is
     exact Submodule.smul_mem _ _ (Submodule.span_subset_span R S _ hy)
 #align is_localization.mem_span_iff IsLocalization.mem_span_iff
 
+set_option synthInstance.etaExperiment true in
 theorem mem_span_map {x : S} {a : Set R} :
     x ∈ Ideal.span (algebraMap R S '' a) ↔ ∃ y ∈ Ideal.span a, ∃ z : M, x = mk' S y z := by
   refine' (mem_span_iff M).trans _
@@ -197,12 +206,14 @@ section CommRing
 
 variable [CommRing K] [Algebra R K] [IsFractionRing R K] [Algebra A K] [IsFractionRing A K]
 
+set_option synthInstance.etaExperiment true in
 @[simp, mono]
 theorem coeSubmodule_le_coeSubmodule {I J : Ideal R} :
     coeSubmodule K I ≤ coeSubmodule K J ↔ I ≤ J :=
   IsLocalization.coeSubmodule_le_coeSubmodule le_rfl
 #align is_fraction_ring.coe_submodule_le_coe_submodule IsFractionRing.coeSubmodule_le_coeSubmodule
 
+set_option synthInstance.etaExperiment true in
 @[mono]
 theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submodule R K) :=
   strictMono_of_le_iff_le fun _ _ => coeSubmodule_le_coeSubmodule.symm
@@ -210,10 +221,12 @@ theorem coeSubmodule_strictMono : StrictMono (coeSubmodule K : Ideal R → Submo
 
 variable (R K)
 
+set_option synthInstance.etaExperiment true in
 theorem coeSubmodule_injective : Function.Injective (coeSubmodule K : Ideal R → Submodule R K) :=
   injective_of_le_imp_le _ fun hl => coeSubmodule_le_coeSubmodule.mp hl
 #align is_fraction_ring.coe_submodule_injective IsFractionRing.coeSubmodule_injective
 
+set_option synthInstance.etaExperiment true in
 @[simp]
 theorem coeSubmodule_isPrincipal {I : Ideal R} : (coeSubmodule K I).IsPrincipal ↔ I.IsPrincipal :=
   IsLocalization.coeSubmodule_isPrincipal _ le_rfl

1ebb2060

feat: port RingTheory.Localization.Submodule (#3514)

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

f0d2fcab

`ring_theory.localization.submodule` ⟷ `Mathlib.RingTheory.Localization.Submodule`

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Important changes

Remaining issues

Slower (failing) search

`simp` not firing sometimes

Missing instances due to unification failing

Workaround for issues

Dependencies 8 + 480

ring_theory.localization.submodule ⟷ Mathlib.RingTheory.Localization.Submodule

Changes since the initial port

Changes in mathlib3

Changes in mathlib3port

Changes in mathlib4

Important changes

Remaining issues

Slower (failing) search

simp not firing sometimes

Missing instances due to unification failing

Workaround for issues

Dependencies 8 + 480

`ring_theory.localization.submodule` ⟷ `Mathlib.RingTheory.Localization.Submodule`

`simp` not firing sometimes