ring_theory.integral_closure ⟷ Mathlib.RingTheory.IntegralClosure

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

@@ -1313,7 +1313,7 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
     have a_inv_ne_zero : a_inv ≠ 0 := right_ne_zero_of_mul (mt ha_inv.symm.trans one_ne_zero)
     refine' (mul_eq_zero.mp _).resolve_right (pow_ne_zero p.nat_degree a_inv_ne_zero)
     rw [eval₂_eq_sum_range] at hp
-    rw [RingHom.map_sum, Finset.sum_mul]
+    rw [map_sum, Finset.sum_mul]
     refine' (Finset.sum_congr rfl fun i hi => _).trans hp
     rw [RingHom.map_mul, mul_assoc]
     congr

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

@@ -3,7 +3,7 @@ Copyright (c) 2019 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 -/
-import Data.Polynomial.Expand
+import Algebra.Polynomial.Expand
 import LinearAlgebra.FiniteDimensional
 import LinearAlgebra.Matrix.Charpoly.LinearMap
 import RingTheory.Adjoin.FG

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

@@ -6,11 +6,11 @@ Authors: Kenny Lau
 import Data.Polynomial.Expand
 import LinearAlgebra.FiniteDimensional
 import LinearAlgebra.Matrix.Charpoly.LinearMap
-import RingTheory.Adjoin.Fg
+import RingTheory.Adjoin.FG
 import RingTheory.FiniteType
 import RingTheory.Polynomial.ScaleRoots
 import RingTheory.Polynomial.Tower
-import RingTheory.TensorProduct
+import LinearAlgebra.TensorProduct.Tower
 
 #align_import ring_theory.integral_closure from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
 
@@ -836,7 +836,7 @@ theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ nat
     MulZeroClass.zero_mul, mem_support_iff, ite_mul, Ne.def, ite_not]
   split_ifs with h₁ h₂
   · simp [h₁]
-  · rw [h₂, leading_coeff, ← pow_succ, tsub_add_cancel_of_le hp]
+  · rw [h₂, leading_coeff, ← pow_succ', tsub_add_cancel_of_le hp]
   · rw [mul_assoc, ← pow_add, tsub_add_cancel_of_le]
     apply Nat.le_pred_of_lt
     rw [lt_iff_le_and_ne]
@@ -854,7 +854,7 @@ theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
   split_ifs with h₁ h₂
   · simp [*]
   · simp [*]
-  · rw [Algebra.id.smul_eq_mul, mul_comm, mul_assoc, ← pow_succ', tsub_right_comm,
+  · rw [Algebra.id.smul_eq_mul, mul_comm, mul_assoc, ← pow_succ, tsub_right_comm,
       tsub_add_cancel_of_le]
     rw [Nat.succ_le_iff]
     exact tsub_pos_of_lt (lt_of_le_of_ne (le_nat_degree_of_ne_zero h₁) h₂)
@@ -1328,7 +1328,7 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
   convert hq using 2
   refine' Finset.sum_congr rfl fun i hi => _
   have : 1 ≤ p.nat_degree - i := le_tsub_of_add_le_left (finset.mem_range.mp hi)
-  rw [mul_assoc, ← pow_succ', tsub_add_cancel_of_le this]
+  rw [mul_assoc, ← pow_succ, tsub_add_cancel_of_le this]
 #align is_field_of_is_integral_of_is_field isField_of_isIntegral_of_isField
 -/
 

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

@@ -108,7 +108,7 @@ theorem isIntegral_of_noetherian (H : IsNoetherian R A) (x : A) : IsIntegral R x
   cases' HM with N HN
   have HM : ¬M < D (N + 1) :=
     WellFounded.not_lt_min (isNoetherian_iff_wellFounded.1 H) (Set.range D) _ ⟨N + 1, rfl⟩
-  rw [← HN] at HM 
+  rw [← HN] at HM
   have HN2 : D (N + 1) ≤ D N :=
     by_contradiction fun H =>
       HM (lt_of_le_not_le (map_mono (degree_le_mono (WithBot.coe_le_coe.2 (Nat.le_succ N)))) H)
@@ -180,7 +180,7 @@ theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Al
   rintro ⟨p, hp, hx⟩
   use p, hp
   rwa [← f.comp_algebra_map, ← AlgHom.coe_toRingHom, ← Polynomial.hom_eval₂, AlgHom.coe_toRingHom,
-    map_eq_zero_iff f hf] at hx 
+    map_eq_zero_iff f hf] at hx
 #align is_integral_alg_hom_iff isIntegral_algHom_iff
 -/
 
@@ -250,14 +250,14 @@ theorem fG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rcases hx with ⟨f, hfm, hfx⟩
   exists Finset.image ((· ^ ·) x) (Finset.range (nat_degree f + 1))
   apply le_antisymm
-  · rw [span_le]; intro s hs; rw [Finset.mem_coe] at hs 
+  · rw [span_le]; intro s hs; rw [Finset.mem_coe] at hs
     rcases Finset.mem_image.1 hs with ⟨k, hk, rfl⟩; clear hk
     exact (Algebra.adjoin R {x}).pow_mem (Algebra.subset_adjoin (Set.mem_singleton _)) k
-  intro r hr; change r ∈ Algebra.adjoin R ({x} : Set A) at hr 
-  rw [Algebra.adjoin_singleton_eq_range_aeval] at hr 
+  intro r hr; change r ∈ Algebra.adjoin R ({x} : Set A) at hr
+  rw [Algebra.adjoin_singleton_eq_range_aeval] at hr
   rcases(aeval x).mem_range.mp hr with ⟨p, rfl⟩
   rw [← mod_by_monic_add_div p hfm]
-  rw [← aeval_def] at hfx 
+  rw [← aeval_def] at hfx
   rw [AlgHom.map_add, AlgHom.map_mul, hfx, MulZeroClass.zero_mul, add_zero]
   have : degree (p %ₘ f) ≤ degree f := degree_mod_by_monic_le p hfm
   generalize p %ₘ f = q at this ⊢
@@ -267,8 +267,8 @@ theorem fG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   refine' smul_mem _ _ (subset_span _)
   rw [Finset.mem_coe]; refine' Finset.mem_image.2 ⟨_, _, rfl⟩
   rw [Finset.mem_range, Nat.lt_succ_iff]; refine' le_of_not_lt fun hk => _
-  rw [degree_le_iff_coeff_zero] at this 
-  rw [mem_support_iff] at hkq ; apply hkq; apply this
+  rw [degree_le_iff_coeff_zero] at this
+  rw [mem_support_iff] at hkq; apply hkq; apply this
   exact lt_of_le_of_lt degree_le_nat_degree (WithBot.coe_lt_coe.2 hk)
 #align fg_adjoin_singleton_of_integral fG_adjoin_singleton_of_integral
 
@@ -319,7 +319,7 @@ theorem IsIntegral.of_mem_of_fg (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   -- Now `S` is a subalgebra so the product of two elements of `y` is also in `S`.
   have : ∀ jk : (↑(y ×ˢ y) : Set (A × A)), jk.1.1 * jk.1.2 ∈ S.to_submodule := fun jk =>
     S.mul_mem (hyS (Finset.mem_product.1 jk.2).1) (hyS (Finset.mem_product.1 jk.2).2)
-  rw [← hy, ← Set.image_id ↑y] at this ; simp only [Finsupp.mem_span_image_iff_total] at this 
+  rw [← hy, ← Set.image_id ↑y] at this; simp only [Finsupp.mem_span_image_iff_total] at this
   -- Say `yᵢyⱼ = ∑rᵢⱼₖ yₖ`
   choose ly hly1 hly2
   -- Now let `S₀` be the subring of `R` generated by the `rᵢ` and the `rᵢⱼₖ`.
@@ -369,7 +369,7 @@ theorem IsIntegral.of_mem_of_fg (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
         (span_le.2
             (Set.insert_subset_iff.2 ⟨(Algebra.adjoin S₀ ↑y).one_mem, Algebra.subset_adjoin⟩))
           hz
-    · rw [Subalgebra.mem_toSubmodule, Algebra.mem_adjoin_iff] at hz 
+    · rw [Subalgebra.mem_toSubmodule, Algebra.mem_adjoin_iff] at hz
       suffices Subring.closure (Set.range ⇑(algebraMap (↥S₀) A) ∪ ↑y) ≤ S₁ by exact this hz
       refine' Subring.closure_le.2 (Set.union_subset _ fun t ht => subset_span <| Or.inr ht)
       rw [Set.range_subset_iff]
@@ -388,7 +388,7 @@ theorem IsIntegral.of_mem_of_fg (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   have : lx r ∈ S₀ :=
     Subring.subset_closure (Finset.mem_union_left _ (Finset.mem_image_of_mem _ hr))
   change (⟨_, this⟩ : S₀) • r ∈ _
-  rw [Finsupp.mem_supported] at hlx1 
+  rw [Finsupp.mem_supported] at hlx1
   exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _
 #align is_integral_of_mem_of_fg IsIntegral.of_mem_of_fg
 -/
@@ -421,7 +421,7 @@ theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R
         map_smul' := fun r s => LinearMap.ext fun n => Subtype.ext <| smul_assoc r s n }
       (LinearMap.ext fun n => Subtype.ext <| one_smul _ _) fun x y =>
       LinearMap.ext fun n => Subtype.ext <| mul_smul x y n
-  obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by by_contra h'; push_neg at h' ; apply hN;
+  obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by by_contra h'; push_neg at h'; apply hN;
     rwa [eq_bot_iff]
   have : Function.Injective f :=
     by
@@ -477,7 +477,7 @@ theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.Fin
   by
   have :=
     h.to_finite (by delta RingHom.FiniteType; convert h'; ext; exact (Algebra.smul_def _ _).symm)
-  delta RingHom.Finite at this ; convert this; ext; exact Algebra.smul_def _ _
+  delta RingHom.Finite at this; convert this; ext; exact Algebra.smul_def _ _
 #align algebra.is_integral.finite Algebra.IsIntegral.finite
 -/
 
@@ -505,7 +505,7 @@ theorem RingHom.IsIntegralElem.of_mem_closure {x y z : S} (hx : f.IsIntegralElem
   by
   letI : Algebra R S := f.to_algebra
   have := (fG_adjoin_singleton_of_integral x hx).mul (fG_adjoin_singleton_of_integral y hy)
-  rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this 
+  rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this
   exact
     IsIntegral.of_mem_of_fg (Algebra.adjoin R {x, y}) this z
       (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)
@@ -692,7 +692,7 @@ theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
   let ⟨p, hpm, hpx⟩ := x.2
   ⟨p, hpm,
     Subtype.eq <| by
-      rwa [← aeval_def, Subtype.val_eq_coe, ← Subalgebra.val_apply, aeval_alg_hom_apply] at hpx ⟩
+      rwa [← aeval_def, Subtype.val_eq_coe, ← Subalgebra.val_apply, aeval_alg_hom_apply] at hpx⟩
 #align integral_closure.is_integral integralClosure.isIntegral
 -/
 
@@ -916,16 +916,16 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
   by
   by_cases h' : 1 ≤ p.nat_degree
   · use normalizeScaleRoots p
-    have : p ≠ 0 := fun h'' => by rw [h'', nat_degree_zero] at h' ; exact Nat.not_succ_le_zero 0 h'
+    have : p ≠ 0 := fun h'' => by rw [h'', nat_degree_zero] at h'; exact Nat.not_succ_le_zero 0 h'
     use normalizeScaleRoots_monic p this
     rw [normalizeScaleRoots_eval₂_leadingCoeff_mul p h' f x, h, MulZeroClass.mul_zero]
   · by_cases hp : p.map f = 0
-    · apply_fun fun q => coeff q p.nat_degree at hp 
-      rw [coeff_map, coeff_zero, coeff_nat_degree] at hp 
+    · apply_fun fun q => coeff q p.nat_degree at hp
+      rw [coeff_map, coeff_zero, coeff_nat_degree] at hp
       rw [hp, MulZeroClass.zero_mul]
       exact f.is_integral_zero
-    · rw [Nat.one_le_iff_ne_zero, Classical.not_not] at h' 
-      rw [eq_C_of_nat_degree_eq_zero h', eval₂_C] at h 
+    · rw [Nat.one_le_iff_ne_zero, Classical.not_not] at h'
+      rw [eq_C_of_nat_degree_eq_zero h', eval₂_C] at h
       suffices p.map f = 0 by exact (hp this).rec _
       rw [eq_C_of_nat_degree_eq_zero h', map_C, h, C_eq_zero]
 #align ring_hom.is_integral_elem_leading_coeff_mul RingHom.isIntegralElem_leadingCoeff_mul
@@ -937,7 +937,7 @@ then `p.leading_coeff • x : S` is integral over `R`. -/
 theorem isIntegral_leadingCoeff_smul [Algebra R S] (h : aeval x p = 0) :
     IsIntegral R (p.leadingCoeff • x) :=
   by
-  rw [aeval_def] at h 
+  rw [aeval_def] at h
   rw [Algebra.smul_def]
   exact (algebraMap R S).isIntegralElem_leadingCoeff_mul p x h
 #align is_integral_leading_coeff_smul isIntegral_leadingCoeff_smul
@@ -1158,7 +1158,7 @@ theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegr
   let S : Set B := ↑(p.map <| algebraMap A B).frange
   refine' IsIntegral.of_mem_of_fg (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
   refine' fg_trans (fg_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
-  · rw [Finset.mem_coe, frange, Finset.mem_image] at hx 
+  · rw [Finset.mem_coe, frange, Finset.mem_image] at hx
     rcases hx with ⟨i, _, rfl⟩
     rw [coeff_map]
     exact IsIntegral.map (IsScalarTower.toAlgHom R A B) (A_int _)
@@ -1207,7 +1207,7 @@ theorem IsIntegral.tower_bot (H : Function.Injective (algebraMap A B)) {x : A}
   rcases h with ⟨p, ⟨hp, hp'⟩⟩
   refine' ⟨p, ⟨hp, _⟩⟩
   rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map, eval₂_hom, ←
-    RingHom.map_zero (algebraMap A B)] at hp' 
+    RingHom.map_zero (algebraMap A B)] at hp'
   rw [eval₂_eq_eval_map]
   exact H hp'
 #align is_integral_tower_bot_of_is_integral IsIntegral.tower_bot
@@ -1235,7 +1235,7 @@ theorem IsIntegral.tower_bot_of_field {R A B : Type _} [CommRing R] [Field A] [C
 theorem RingHom.isIntegralElem.of_comp {x : T} (h : (g.comp f).IsIntegralElem x) :
     g.IsIntegralElem x :=
   let ⟨p, ⟨hp, hp'⟩⟩ := h
-  ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp' ⟩
+  ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp'⟩
 #align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem.of_comp
 -/
 
@@ -1254,7 +1254,7 @@ theorem IsIntegral.tower_top {x : B} (h : IsIntegral R x) : IsIntegral A x :=
   by
   rcases h with ⟨p, ⟨hp, hp'⟩⟩
   refine' ⟨p.map (algebraMap R A), ⟨hp.map (algebraMap R A), _⟩⟩
-  rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map] at hp' 
+  rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map] at hp'
   exact hp'
 #align is_integral_tower_top_of_is_integral IsIntegral.tower_top
 -/
@@ -1312,7 +1312,7 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
     apply (injective_iff_map_eq_zero (algebraMap R S)).mp hRS
     have a_inv_ne_zero : a_inv ≠ 0 := right_ne_zero_of_mul (mt ha_inv.symm.trans one_ne_zero)
     refine' (mul_eq_zero.mp _).resolve_right (pow_ne_zero p.nat_degree a_inv_ne_zero)
-    rw [eval₂_eq_sum_range] at hp 
+    rw [eval₂_eq_sum_range] at hp
     rw [RingHom.map_sum, Finset.sum_mul]
     refine' (Finset.sum_congr rfl fun i hi => _).trans hp
     rw [RingHom.map_mul, mul_assoc]
@@ -1323,7 +1323,7 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
   -- Since `q(a) = 0` and `q(a) = q'(a) * a + 1`, we have `a * -q'(a) = 1`.
   -- TODO: we could use a lemma for `polynomial.div_X` here.
   rw [Finset.sum_range_succ_comm, p_monic.coeff_nat_degree, one_mul, tsub_self, pow_zero,
-    add_eq_zero_iff_eq_neg, eq_comm] at hq 
+    add_eq_zero_iff_eq_neg, eq_comm] at hq
   rw [mul_comm, neg_mul, Finset.sum_mul]
   convert hq using 2
   refine' Finset.sum_congr rfl fun i hi => _

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

@@ -950,7 +950,7 @@ end
 section IsIntegralClosure
 
 #print IsIntegralClosure /-
-/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:400:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
 /-- `is_integral_closure A R B` is the characteristic predicate stating `A` is
 the integral closure of `R` in `B`,
 i.e. that an element of `B` is integral over `R` iff it is an element of (the image of) `A`.

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

@@ -950,7 +950,7 @@ end
 section IsIntegralClosure
 
 #print IsIntegralClosure /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:404:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
 /-- `is_integral_closure A R B` is the characteristic predicate stating `A` is
 the integral closure of `R` in `B`,
 i.e. that an element of `B` is integral over `R` iff it is an element of (the image of) `A`.

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

@@ -244,8 +244,7 @@ theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
 #align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finite
 -/
 
-#print IsIntegral.fg_adjoin_singleton /-
-theorem IsIntegral.fg_adjoin_singleton (x : A) (hx : IsIntegral R x) :
+theorem fG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
     (Algebra.adjoin R ({x} : Set A)).toSubmodule.FG :=
   by
   rcases hx with ⟨f, hfm, hfx⟩
@@ -271,8 +270,7 @@ theorem IsIntegral.fg_adjoin_singleton (x : A) (hx : IsIntegral R x) :
   rw [degree_le_iff_coeff_zero] at this 
   rw [mem_support_iff] at hkq ; apply hkq; apply this
   exact lt_of_le_of_lt degree_le_nat_degree (WithBot.coe_lt_coe.2 hk)
-#align fg_adjoin_singleton_of_integral IsIntegral.fg_adjoin_singleton
--/
+#align fg_adjoin_singleton_of_integral fG_adjoin_singleton_of_integral
 
 #print fg_adjoin_of_finite /-
 theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
@@ -289,7 +287,7 @@ theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
       rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule] <;>
         exact
           fg.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)
-            (IsIntegral.fg_adjoin_singleton _ <| his a <| Set.mem_insert a s))
+            (fG_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s))
     his
 #align fg_adjoin_of_finite fg_adjoin_of_finite
 -/
@@ -506,7 +504,7 @@ theorem RingHom.IsIntegralElem.of_mem_closure {x y z : S} (hx : f.IsIntegralElem
     (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z :=
   by
   letI : Algebra R S := f.to_algebra
-  have := (IsIntegral.fg_adjoin_singleton x hx).mul (IsIntegral.fg_adjoin_singleton y hy)
+  have := (fG_adjoin_singleton_of_integral x hx).mul (fG_adjoin_singleton_of_integral y hy)
   rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this 
   exact
     IsIntegral.of_mem_of_fg (Algebra.adjoin R {x, y}) this z
@@ -638,7 +636,8 @@ def integralClosure : Subalgebra R A
 #print mem_integralClosure_iff_mem_fg /-
 theorem mem_integralClosure_iff_mem_fg {r : A} :
     r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M :=
-  ⟨fun hr => ⟨Algebra.adjoin R {r}, IsIntegral.fg_adjoin_singleton _ hr, Algebra.subset_adjoin rfl⟩,
+  ⟨fun hr =>
+    ⟨Algebra.adjoin R {r}, fG_adjoin_singleton_of_integral _ hr, Algebra.subset_adjoin rfl⟩,
     fun ⟨M, Hf, hrM⟩ => IsIntegral.of_mem_of_fg M Hf _ hrM⟩
 #align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_fg
 -/
@@ -1163,7 +1162,7 @@ theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegr
     rcases hx with ⟨i, _, rfl⟩
     rw [coeff_map]
     exact IsIntegral.map (IsScalarTower.toAlgHom R A B) (A_int _)
-  · apply IsIntegral.fg_adjoin_singleton
+  · apply fG_adjoin_singleton_of_integral
     exact isIntegral_trans_aux _ pmonic hp
 #align is_integral_trans isIntegral_trans
 -/
@@ -1341,7 +1340,7 @@ theorem isField_of_isIntegral_of_isField' {R S : Type _} [CommRing R] [CommRing
   refine' ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun x hx => _⟩
   let A := Algebra.adjoin R ({x} : Set S)
   haveI : IsNoetherian R A :=
-    isNoetherian_of_fg_of_noetherian A.to_submodule (IsIntegral.fg_adjoin_singleton x (H x))
+    isNoetherian_of_fg_of_noetherian A.to_submodule (fG_adjoin_singleton_of_integral x (H x))
   haveI : Module.Finite R A := Module.IsNoetherian.finite R A
   obtain ⟨y, hy⟩ :=
     LinearMap.surjective_of_injective

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

@@ -86,15 +86,15 @@ protected def Algebra.IsIntegral : Prop :=
 
 variable {R A}
 
-#print RingHom.is_integral_map /-
-theorem RingHom.is_integral_map {x : R} : f.IsIntegralElem (f x) :=
+#print RingHom.isIntegralElem_map /-
+theorem RingHom.isIntegralElem_map {x : R} : f.IsIntegralElem (f x) :=
   ⟨X - C x, monic_X_sub_C _, by simp⟩
-#align ring_hom.is_integral_map RingHom.is_integral_map
+#align ring_hom.is_integral_map RingHom.isIntegralElem_map
 -/
 
 #print isIntegral_algebraMap /-
 theorem isIntegral_algebraMap {x : R} : IsIntegral R (algebraMap R A x) :=
-  (algebraMap R A).is_integral_map
+  (algebraMap R A).isIntegralElem_map
 #align is_integral_algebra_map isIntegral_algebraMap
 -/
 
@@ -147,8 +147,8 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S]
 
 variable [Algebra R A] [Algebra R B] (f : R →+* S)
 
-#print map_isIntegral /-
-theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
+#print IsIntegral.map /-
+theorem IsIntegral.map {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
     [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B} [AlgHomClass F A B C] (f : F)
     (hb : IsIntegral R b) : IsIntegral R (f b) :=
   by
@@ -156,12 +156,12 @@ theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra
   refine' ⟨P, hP.1, _⟩
   rw [← aeval_def, show (aeval (f b)) P = (aeval (f b)) (P.map (algebraMap R A)) by simp,
     aeval_alg_hom_apply, aeval_map_algebra_map, aeval_def, hP.2, _root_.map_zero]
-#align map_is_integral map_isIntegral
+#align map_is_integral IsIntegral.map
 -/
 
-#print isIntegral_map_of_comp_eq_of_isIntegral /-
-theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R] [CommRing S]
-    [CommRing T] [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
+#print IsIntegral.map_of_comp_eq /-
+theorem IsIntegral.map_of_comp_eq {R S T U : Type _} [CommRing R] [CommRing S] [CommRing T]
+    [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
     (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
     IsIntegral T (ψ a) := by
   rw [IsIntegral, RingHom.IsIntegralElem] at ha ⊢
@@ -169,14 +169,14 @@ theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R]
   refine' ⟨p.map φ, hp.left.map _, _⟩
   rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply, eval_map, hp.right,
     RingHom.map_zero]
-#align is_integral_map_of_comp_eq_of_is_integral isIntegral_map_of_comp_eq_of_isIntegral
+#align is_integral_map_of_comp_eq_of_is_integral IsIntegral.map_of_comp_eq
 -/
 
 #print isIntegral_algHom_iff /-
 theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A →ₐ[R] B) (hf : Function.Injective f) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
   by
-  refine' ⟨_, map_isIntegral f⟩
+  refine' ⟨_, IsIntegral.map f⟩
   rintro ⟨p, hp, hx⟩
   use p, hp
   rwa [← f.comp_algebra_map, ← AlgHom.coe_toRingHom, ← Polynomial.hom_eval₂, AlgHom.coe_toRingHom,
@@ -188,29 +188,29 @@ theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Al
 @[simp]
 theorem isIntegral_algEquiv {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
-  ⟨fun h => by simpa using map_isIntegral f.symm.to_alg_hom h, map_isIntegral f.toAlgHom⟩
+  ⟨fun h => by simpa using IsIntegral.map f.symm.to_alg_hom h, IsIntegral.map f.toAlgHom⟩
 #align is_integral_alg_equiv isIntegral_algEquiv
 -/
 
-#print isIntegral_of_isScalarTower /-
-theorem isIntegral_of_isScalarTower [Algebra A B] [IsScalarTower R A B] {x : B}
-    (hx : IsIntegral R x) : IsIntegral A x :=
+#print IsIntegral.tower_top /-
+theorem IsIntegral.tower_top [Algebra A B] [IsScalarTower R A B] {x : B} (hx : IsIntegral R x) :
+    IsIntegral A x :=
   let ⟨p, hp, hpx⟩ := hx
   ⟨p.map <| algebraMap R A, hp.map _, by rw [← aeval_def, aeval_map_algebra_map, aeval_def, hpx]⟩
-#align is_integral_of_is_scalar_tower isIntegral_of_isScalarTower
+#align is_integral_of_is_scalar_tower IsIntegral.tower_top
 -/
 
 #print map_isIntegral_int /-
 theorem map_isIntegral_int {B C F : Type _} [Ring B] [Ring C] {b : B} [RingHomClass F B C] (f : F)
     (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
-  map_isIntegral (f : B →+* C).toIntAlgHom hb
+  IsIntegral.map (f : B →+* C).toIntAlgHom hb
 #align map_is_integral_int map_isIntegral_int
 -/
 
-#print isIntegral_ofSubring /-
-theorem isIntegral_ofSubring {x : A} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
-  isIntegral_of_isScalarTower hx
-#align is_integral_of_subring isIntegral_ofSubring
+#print IsIntegral.of_subring /-
+theorem IsIntegral.of_subring {x : A} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
+  IsIntegral.tower_top hx
+#align is_integral_of_subring IsIntegral.of_subring
 -/
 
 #print IsIntegral.algebraMap /-
@@ -240,12 +240,12 @@ theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
     refine' ⟨_, Finset.finite_toSet _, p.restriction, monic_restriction.2 hmp, _⟩
     rw [← aeval_def, ← aeval_map_algebra_map R r p.restriction, map_restriction, aeval_def, hpr]
   rcases hr with ⟨s, hs, hsr⟩
-  exact isIntegral_ofSubring _ hsr
+  exact IsIntegral.of_subring _ hsr
 #align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finite
 -/
 
-#print FG_adjoin_singleton_of_integral /-
-theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
+#print IsIntegral.fg_adjoin_singleton /-
+theorem IsIntegral.fg_adjoin_singleton (x : A) (hx : IsIntegral R x) :
     (Algebra.adjoin R ({x} : Set A)).toSubmodule.FG :=
   by
   rcases hx with ⟨f, hfm, hfx⟩
@@ -271,11 +271,11 @@ theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rw [degree_le_iff_coeff_zero] at this 
   rw [mem_support_iff] at hkq ; apply hkq; apply this
   exact lt_of_le_of_lt degree_le_nat_degree (WithBot.coe_lt_coe.2 hk)
-#align fg_adjoin_singleton_of_integral FG_adjoin_singleton_of_integral
+#align fg_adjoin_singleton_of_integral IsIntegral.fg_adjoin_singleton
 -/
 
-#print FG_adjoin_of_finite /-
-theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
+#print fg_adjoin_of_finite /-
+theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
     (Algebra.adjoin R s).toSubmodule.FG :=
   Set.Finite.induction_on hfs
     (fun _ =>
@@ -289,23 +289,23 @@ theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
       rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule] <;>
         exact
           fg.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)
-            (FG_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s))
+            (IsIntegral.fg_adjoin_singleton _ <| his a <| Set.mem_insert a s))
     his
-#align fg_adjoin_of_finite FG_adjoin_of_finite
+#align fg_adjoin_of_finite fg_adjoin_of_finite
 -/
 
 #print isNoetherian_adjoin_finset /-
 theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
     (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (↑s : Set A)) :=
-  isNoetherian_of_fg_of_noetherian _ (FG_adjoin_of_finite s.finite_toSet hs)
+  isNoetherian_of_fg_of_noetherian _ (fg_adjoin_of_finite s.finite_toSet hs)
 #align is_noetherian_adjoin_finset isNoetherian_adjoin_finset
 -/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-#print isIntegral_of_mem_of_FG /-
+#print IsIntegral.of_mem_of_fg /-
 /-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
   then all elements of `S` are integral over `R`. -/
-theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) :
+theorem IsIntegral.of_mem_of_fg (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) :
     IsIntegral R x :=
   by
   -- say `x ∈ S`. We want to prove that `x` is integral over `R`.
@@ -328,7 +328,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   let S₀ : Subring R :=
     Subring.closure ↑(lx.frange ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))
   -- It suffices to prove that `x` is integral over `S₀`.
-  refine' isIntegral_ofSubring S₀ _
+  refine' IsIntegral.of_subring S₀ _
   letI : CommRing S₀ := SubringClass.toCommRing S₀
   letI : Algebra S₀ A := Algebra.ofSubring S₀
   -- Claim: the `S₀`-module span (in `A`) of the set `y ∪ {1}` is closed under
@@ -392,7 +392,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   change (⟨_, this⟩ : S₀) • r ∈ _
   rw [Finsupp.mem_supported] at hlx1 
   exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _
-#align is_integral_of_mem_of_fg isIntegral_of_mem_of_FG
+#align is_integral_of_mem_of_fg IsIntegral.of_mem_of_fg
 -/
 
 #print Module.End.isIntegral /-
@@ -444,7 +444,7 @@ variable {f}
 #print RingHom.Finite.to_isIntegral /-
 theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
   letI := f.to_algebra
-  fun x => isIntegral_of_mem_of_FG ⊤ h.1 _ trivial
+  fun x => IsIntegral.of_mem_of_fg ⊤ h.1 _ trivial
 #align ring_hom.finite.to_is_integral RingHom.Finite.to_isIntegral
 -/
 
@@ -459,7 +459,7 @@ theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.
   constructor
   change (⊤ : Subalgebra R S).toSubmodule.FG
   rw [← hs]
-  exact FG_adjoin_of_finite (Set.toFinite _) fun x _ => h x
+  exact fg_adjoin_of_finite (Set.toFinite _) fun x _ => h x
 #align ring_hom.is_integral.to_finite RingHom.IsIntegral.to_finite
 -/
 
@@ -501,123 +501,123 @@ theorem Algebra.finite_iff_isIntegral_and_finiteType :
 
 variable (f)
 
-#print RingHom.is_integral_of_mem_closure /-
-theorem RingHom.is_integral_of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
+#print RingHom.IsIntegralElem.of_mem_closure /-
+theorem RingHom.IsIntegralElem.of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
     (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z :=
   by
   letI : Algebra R S := f.to_algebra
-  have := (FG_adjoin_singleton_of_integral x hx).mul (FG_adjoin_singleton_of_integral y hy)
+  have := (IsIntegral.fg_adjoin_singleton x hx).mul (IsIntegral.fg_adjoin_singleton y hy)
   rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this 
   exact
-    isIntegral_of_mem_of_FG (Algebra.adjoin R {x, y}) this z
+    IsIntegral.of_mem_of_fg (Algebra.adjoin R {x, y}) this z
       (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)
-#align ring_hom.is_integral_of_mem_closure RingHom.is_integral_of_mem_closure
+#align ring_hom.is_integral_of_mem_closure RingHom.IsIntegralElem.of_mem_closure
 -/
 
-#print isIntegral_of_mem_closure /-
-theorem isIntegral_of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
+#print IsIntegral.of_mem_closure /-
+theorem IsIntegral.of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
     (hz : z ∈ Subring.closure ({x, y} : Set A)) : IsIntegral R z :=
-  (algebraMap R A).is_integral_of_mem_closure hx hy hz
-#align is_integral_of_mem_closure isIntegral_of_mem_closure
+  (algebraMap R A).of_mem_closure hx hy hz
+#align is_integral_of_mem_closure IsIntegral.of_mem_closure
 -/
 
-#print RingHom.is_integral_zero /-
-theorem RingHom.is_integral_zero : f.IsIntegralElem 0 :=
-  f.map_zero ▸ f.is_integral_map
-#align ring_hom.is_integral_zero RingHom.is_integral_zero
+#print RingHom.isIntegralElem_zero /-
+theorem RingHom.isIntegralElem_zero : f.IsIntegralElem 0 :=
+  f.map_zero ▸ f.isIntegralElem_map
+#align ring_hom.is_integral_zero RingHom.isIntegralElem_zero
 -/
 
 #print isIntegral_zero /-
 theorem isIntegral_zero : IsIntegral R (0 : A) :=
-  (algebraMap R A).is_integral_zero
+  (algebraMap R A).isIntegralElem_zero
 #align is_integral_zero isIntegral_zero
 -/
 
-#print RingHom.is_integral_one /-
-theorem RingHom.is_integral_one : f.IsIntegralElem 1 :=
-  f.map_one ▸ f.is_integral_map
-#align ring_hom.is_integral_one RingHom.is_integral_one
+#print RingHom.isIntegralElem_one /-
+theorem RingHom.isIntegralElem_one : f.IsIntegralElem 1 :=
+  f.map_one ▸ f.isIntegralElem_map
+#align ring_hom.is_integral_one RingHom.isIntegralElem_one
 -/
 
 #print isIntegral_one /-
 theorem isIntegral_one : IsIntegral R (1 : A) :=
-  (algebraMap R A).is_integral_one
+  (algebraMap R A).isIntegralElem_one
 #align is_integral_one isIntegral_one
 -/
 
-#print RingHom.is_integral_add /-
-theorem RingHom.is_integral_add {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
+#print RingHom.IsIntegralElem.add /-
+theorem RingHom.IsIntegralElem.add {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x + y) :=
-  f.is_integral_of_mem_closure hx hy <|
+  f.of_mem_closure hx hy <|
     Subring.add_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl))
-#align ring_hom.is_integral_add RingHom.is_integral_add
+#align ring_hom.is_integral_add RingHom.IsIntegralElem.add
 -/
 
-#print isIntegral_add /-
-theorem isIntegral_add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
+#print IsIntegral.add /-
+theorem IsIntegral.add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x + y) :=
-  (algebraMap R A).is_integral_add hx hy
-#align is_integral_add isIntegral_add
+  (algebraMap R A).add hx hy
+#align is_integral_add IsIntegral.add
 -/
 
-#print RingHom.is_integral_neg /-
-theorem RingHom.is_integral_neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) :=
-  f.is_integral_of_mem_closure hx hx (Subring.neg_mem _ (Subring.subset_closure (Or.inl rfl)))
-#align ring_hom.is_integral_neg RingHom.is_integral_neg
+#print RingHom.IsIntegralElem.neg /-
+theorem RingHom.IsIntegralElem.neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) :=
+  f.of_mem_closure hx hx (Subring.neg_mem _ (Subring.subset_closure (Or.inl rfl)))
+#align ring_hom.is_integral_neg RingHom.IsIntegralElem.neg
 -/
 
-#print isIntegral_neg /-
-theorem isIntegral_neg {x : A} (hx : IsIntegral R x) : IsIntegral R (-x) :=
-  (algebraMap R A).is_integral_neg hx
-#align is_integral_neg isIntegral_neg
+#print IsIntegral.neg /-
+theorem IsIntegral.neg {x : A} (hx : IsIntegral R x) : IsIntegral R (-x) :=
+  (algebraMap R A).neg hx
+#align is_integral_neg IsIntegral.neg
 -/
 
-#print RingHom.is_integral_sub /-
-theorem RingHom.is_integral_sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
+#print RingHom.IsIntegralElem.sub /-
+theorem RingHom.IsIntegralElem.sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x - y) := by
   simpa only [sub_eq_add_neg] using f.is_integral_add hx (f.is_integral_neg hy)
-#align ring_hom.is_integral_sub RingHom.is_integral_sub
+#align ring_hom.is_integral_sub RingHom.IsIntegralElem.sub
 -/
 
-#print isIntegral_sub /-
-theorem isIntegral_sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
+#print IsIntegral.sub /-
+theorem IsIntegral.sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x - y) :=
-  (algebraMap R A).is_integral_sub hx hy
-#align is_integral_sub isIntegral_sub
+  (algebraMap R A).sub hx hy
+#align is_integral_sub IsIntegral.sub
 -/
 
-#print RingHom.is_integral_mul /-
-theorem RingHom.is_integral_mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
+#print RingHom.IsIntegralElem.mul /-
+theorem RingHom.IsIntegralElem.mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x * y) :=
-  f.is_integral_of_mem_closure hx hy
+  f.of_mem_closure hx hy
     (Subring.mul_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl)))
-#align ring_hom.is_integral_mul RingHom.is_integral_mul
+#align ring_hom.is_integral_mul RingHom.IsIntegralElem.mul
 -/
 
-#print isIntegral_mul /-
-theorem isIntegral_mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
+#print IsIntegral.mul /-
+theorem IsIntegral.mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x * y) :=
-  (algebraMap R A).is_integral_mul hx hy
-#align is_integral_mul isIntegral_mul
+  (algebraMap R A).mul hx hy
+#align is_integral_mul IsIntegral.mul
 -/
 
-#print isIntegral_smul /-
-theorem isIntegral_smul [Algebra S A] [Algebra R S] [IsScalarTower R S A] {x : A} (r : R)
+#print IsIntegral.smul /-
+theorem IsIntegral.smul [Algebra S A] [Algebra R S] [IsScalarTower R S A] {x : A} (r : R)
     (hx : IsIntegral S x) : IsIntegral S (r • x) :=
   by
   rw [Algebra.smul_def, IsScalarTower.algebraMap_apply R S A]
-  exact isIntegral_mul isIntegral_algebraMap hx
-#align is_integral_smul isIntegral_smul
+  exact IsIntegral.mul isIntegral_algebraMap hx
+#align is_integral_smul IsIntegral.smul
 -/
 
-#print isIntegral_of_pow /-
-theorem isIntegral_of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
+#print IsIntegral.of_pow /-
+theorem IsIntegral.of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
     IsIntegral R x := by
   rcases hx with ⟨p, ⟨hmonic, heval⟩⟩
   exact
     ⟨expand R n p, monic.expand hn hmonic, by
       rwa [eval₂_eq_eval_map, map_expand, expand_eval, ← eval₂_eq_eval_map]⟩
-#align is_integral_of_pow isIntegral_of_pow
+#align is_integral_of_pow IsIntegral.of_pow
 -/
 
 variable (R A)
@@ -629,19 +629,18 @@ def integralClosure : Subalgebra R A
   carrier := {r | IsIntegral R r}
   zero_mem' := isIntegral_zero
   one_mem' := isIntegral_one
-  add_mem' _ _ := isIntegral_add
-  hMul_mem' _ _ := isIntegral_mul
+  add_mem' _ _ := IsIntegral.add
+  hMul_mem' _ _ := IsIntegral.mul
   algebraMap_mem' x := isIntegral_algebraMap
 #align integral_closure integralClosure
 -/
 
-#print mem_integralClosure_iff_mem_FG /-
-theorem mem_integralClosure_iff_mem_FG {r : A} :
+#print mem_integralClosure_iff_mem_fg /-
+theorem mem_integralClosure_iff_mem_fg {r : A} :
     r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M :=
-  ⟨fun hr =>
-    ⟨Algebra.adjoin R {r}, FG_adjoin_singleton_of_integral _ hr, Algebra.subset_adjoin rfl⟩,
-    fun ⟨M, Hf, hrM⟩ => isIntegral_of_mem_of_FG M Hf _ hrM⟩
-#align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_FG
+  ⟨fun hr => ⟨Algebra.adjoin R {r}, IsIntegral.fg_adjoin_singleton _ hr, Algebra.subset_adjoin rfl⟩,
+    fun ⟨M, Hf, hrM⟩ => IsIntegral.of_mem_of_fg M Hf _ hrM⟩
+#align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_fg
 -/
 
 variable {R} {A}
@@ -666,11 +665,11 @@ theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
 #align le_integral_closure_iff_is_integral le_integralClosure_iff_isIntegral
 -/
 
-#print isIntegral_sup /-
-theorem isIntegral_sup {S T : Subalgebra R A} :
+#print Algebra.isIntegral_sup /-
+theorem Algebra.isIntegral_sup {S T : Subalgebra R A} :
     Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R S ∧ Algebra.IsIntegral R T := by
   simp only [← le_integralClosure_iff_isIntegral, sup_le_iff]
-#align is_integral_sup isIntegral_sup
+#align is_integral_sup Algebra.isIntegral_sup
 -/
 
 #print integralClosure_map_algEquiv /-
@@ -682,9 +681,9 @@ theorem integralClosure_map_algEquiv (f : A ≃ₐ[R] B) :
   rw [Subalgebra.mem_map]
   constructor
   · rintro ⟨x, hx, rfl⟩
-    exact map_isIntegral f hx
+    exact IsIntegral.map f hx
   · intro hy
-    use f.symm y, map_isIntegral (f.symm : B →ₐ[R] A) hy
+    use f.symm y, IsIntegral.map (f.symm : B →ₐ[R] A) hy
     simp
 #align integral_closure_map_alg_equiv integralClosure_map_algEquiv
 -/
@@ -698,38 +697,38 @@ theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
 #align integral_closure.is_integral integralClosure.isIntegral
 -/
 
-#print RingHom.isIntegral_of_isIntegral_mul_unit /-
-theorem RingHom.isIntegral_of_isIntegral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
+#print RingHom.IsIntegralElem.of_mul_unit /-
+theorem RingHom.IsIntegralElem.of_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
     (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x :=
   by
   obtain ⟨p, ⟨p_monic, hp⟩⟩ := hx
   refine' ⟨scale_roots p r, ⟨(monic_scale_roots_iff r).2 p_monic, _⟩⟩
   convert scale_roots_eval₂_eq_zero f hp
   rw [mul_comm x y, ← mul_assoc, hr, one_mul]
-#align ring_hom.is_integral_of_is_integral_mul_unit RingHom.isIntegral_of_isIntegral_mul_unit
+#align ring_hom.is_integral_of_is_integral_mul_unit RingHom.IsIntegralElem.of_mul_unit
 -/
 
-#print isIntegral_of_isIntegral_mul_unit /-
-theorem isIntegral_of_isIntegral_mul_unit {x y : A} {r : R} (hr : algebraMap R A r * y = 1)
+#print IsIntegral.of_mul_unit /-
+theorem IsIntegral.of_mul_unit {x y : A} {r : R} (hr : algebraMap R A r * y = 1)
     (hx : IsIntegral R (x * y)) : IsIntegral R x :=
-  (algebraMap R A).isIntegral_of_isIntegral_mul_unit x y r hr hx
-#align is_integral_of_is_integral_mul_unit isIntegral_of_isIntegral_mul_unit
+  (algebraMap R A).of_mul_unit x y r hr hx
+#align is_integral_of_is_integral_mul_unit IsIntegral.of_mul_unit
 -/
 
-#print isIntegral_of_mem_closure' /-
+#print IsIntegral.of_mem_closure' /-
 /-- Generalization of `is_integral_of_mem_closure` bootstrapped up from that lemma -/
-theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) :
+theorem IsIntegral.of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) :
     ∀ x ∈ Subring.closure G, IsIntegral R x := fun x hx =>
-  Subring.closure_induction hx hG isIntegral_zero isIntegral_one (fun _ _ => isIntegral_add)
-    (fun _ => isIntegral_neg) fun _ _ => isIntegral_mul
-#align is_integral_of_mem_closure' isIntegral_of_mem_closure'
+  Subring.closure_induction hx hG isIntegral_zero isIntegral_one (fun _ _ => IsIntegral.add)
+    (fun _ => IsIntegral.neg) fun _ _ => IsIntegral.mul
+#align is_integral_of_mem_closure' IsIntegral.of_mem_closure'
 -/
 
-#print isIntegral_of_mem_closure'' /-
-theorem isIntegral_of_mem_closure'' {S : Type _} [CommRing S] {f : R →+* S} (G : Set S)
+#print IsIntegral.of_mem_closure'' /-
+theorem IsIntegral.of_mem_closure'' {S : Type _} [CommRing S] {f : R →+* S} (G : Set S)
     (hG : ∀ x ∈ G, f.IsIntegralElem x) : ∀ x ∈ Subring.closure G, f.IsIntegralElem x := fun x hx =>
-  @isIntegral_of_mem_closure' R S _ _ f.toAlgebra G hG x hx
-#align is_integral_of_mem_closure'' isIntegral_of_mem_closure''
+  @IsIntegral.of_mem_closure' R S _ _ f.toAlgebra G hG x hx
+#align is_integral_of_mem_closure'' IsIntegral.of_mem_closure''
 -/
 
 #print IsIntegral.pow /-
@@ -790,7 +789,7 @@ theorem IsIntegral.det {n : Type _} [Fintype n] [DecidableEq n] {M : Matrix n n
 #print IsIntegral.pow_iff /-
 @[simp]
 theorem IsIntegral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n) ↔ IsIntegral R x :=
-  ⟨isIntegral_of_pow hn, fun hx => IsIntegral.pow hx n⟩
+  ⟨IsIntegral.of_pow hn, fun hx => IsIntegral.pow hx n⟩
 #align is_integral.pow_iff IsIntegral.pow_iff
 -/
 
@@ -1036,7 +1035,7 @@ theorem mk'_zero (h : IsIntegral R (0 : B) := isIntegral_zero) : mk' A 0 h = 0 :
 #print IsIntegralClosure.mk'_add /-
 @[simp]
 theorem mk'_add (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
-    mk' A (x + y) (isIntegral_add hx hy) = mk' A x hx + mk' A y hy :=
+    mk' A (x + y) (IsIntegral.add hx hy) = mk' A x hx + mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebra_map_mk', RingHom.map_add]
 #align is_integral_closure.mk'_add IsIntegralClosure.mk'_add
 -/
@@ -1044,7 +1043,7 @@ theorem mk'_add (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
 #print IsIntegralClosure.mk'_mul /-
 @[simp]
 theorem mk'_mul (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
-    mk' A (x * y) (isIntegral_mul hx hy) = mk' A x hx * mk' A y hy :=
+    mk' A (x * y) (IsIntegral.mul hx hy) = mk' A x hx * mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebra_map_mk', RingHom.map_mul]
 #align is_integral_closure.mk'_mul IsIntegralClosure.mk'_mul
 -/
@@ -1125,7 +1124,6 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S] [CommRing T]
 
 variable [Algebra A B] [Algebra R B] (f : R →+* S) (g : S →+* T)
 
-#print isIntegral_trans_aux /-
 theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x p = 0) :
     IsIntegral (adjoin R (↑(p.map <| algebraMap A B).frange : Set B)) x :=
   by
@@ -1149,7 +1147,6 @@ theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x
     replace hq := congr_arg (eval x) hq
     convert hq using 1 <;> symm <;> apply eval_map
 #align is_integral_trans_aux isIntegral_trans_aux
--/
 
 variable [Algebra R A] [IsScalarTower R A B]
 
@@ -1160,52 +1157,52 @@ theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegr
     IsIntegral R x := by
   rcases hx with ⟨p, pmonic, hp⟩
   let S : Set B := ↑(p.map <| algebraMap A B).frange
-  refine' isIntegral_of_mem_of_FG (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
-  refine' fg_trans (FG_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
+  refine' IsIntegral.of_mem_of_fg (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
+  refine' fg_trans (fg_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
   · rw [Finset.mem_coe, frange, Finset.mem_image] at hx 
     rcases hx with ⟨i, _, rfl⟩
     rw [coeff_map]
-    exact map_isIntegral (IsScalarTower.toAlgHom R A B) (A_int _)
-  · apply FG_adjoin_singleton_of_integral
+    exact IsIntegral.map (IsScalarTower.toAlgHom R A B) (A_int _)
+  · apply IsIntegral.fg_adjoin_singleton
     exact isIntegral_trans_aux _ pmonic hp
 #align is_integral_trans isIntegral_trans
 -/
 
-#print Algebra.isIntegral_trans /-
+#print Algebra.IsIntegral.trans /-
 /-- If A is an R-algebra all of whose elements are integral over R,
 and B is an A-algebra all of whose elements are integral over A,
 then all elements of B are integral over R.-/
-theorem Algebra.isIntegral_trans (hA : Algebra.IsIntegral R A) (hB : Algebra.IsIntegral A B) :
+theorem Algebra.IsIntegral.trans (hA : Algebra.IsIntegral R A) (hB : Algebra.IsIntegral A B) :
     Algebra.IsIntegral R B := fun x => isIntegral_trans hA x (hB x)
-#align algebra.is_integral_trans Algebra.isIntegral_trans
+#align algebra.is_integral_trans Algebra.IsIntegral.trans
 -/
 
-#print RingHom.isIntegral_trans /-
-theorem RingHom.isIntegral_trans (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.comp f).IsIntegral :=
-  @Algebra.isIntegral_trans R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
+#print RingHom.IsIntegral.trans /-
+theorem RingHom.IsIntegral.trans (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.comp f).IsIntegral :=
+  @Algebra.IsIntegral.trans R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
     (@IsScalarTower.of_algebraMap_eq R S T _ _ _ f.toAlgebra g.toAlgebra (g.comp f).toAlgebra
       (RingHom.comp_apply g f))
     hf hg
-#align ring_hom.is_integral_trans RingHom.isIntegral_trans
+#align ring_hom.is_integral_trans RingHom.IsIntegral.trans
 -/
 
 #print RingHom.isIntegral_of_surjective /-
 theorem RingHom.isIntegral_of_surjective (hf : Function.Surjective f) : f.IsIntegral := fun x =>
-  (hf x).recOn fun y hy => (hy ▸ f.is_integral_map : f.IsIntegralElem x)
+  (hf x).recOn fun y hy => (hy ▸ f.isIntegralElem_map : f.IsIntegralElem x)
 #align ring_hom.is_integral_of_surjective RingHom.isIntegral_of_surjective
 -/
 
-#print isIntegral_of_surjective /-
-theorem isIntegral_of_surjective (h : Function.Surjective (algebraMap R A)) :
+#print Algebra.isIntegral_of_surjective /-
+theorem Algebra.isIntegral_of_surjective (h : Function.Surjective (algebraMap R A)) :
     Algebra.IsIntegral R A :=
   (algebraMap R A).isIntegral_of_surjective h
-#align is_integral_of_surjective isIntegral_of_surjective
+#align is_integral_of_surjective Algebra.isIntegral_of_surjective
 -/
 
-#print isIntegral_tower_bot_of_isIntegral /-
+#print IsIntegral.tower_bot /-
 /-- If `R → A → B` is an algebra tower with `A → B` injective,
 then if the entire tower is an integral extension so is `R → A` -/
-theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A B)) {x : A}
+theorem IsIntegral.tower_bot (H : Function.Injective (algebraMap A B)) {x : A}
     (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
   by
   rcases h with ⟨p, ⟨hp, hp'⟩⟩
@@ -1214,69 +1211,71 @@ theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A
     RingHom.map_zero (algebraMap A B)] at hp' 
   rw [eval₂_eq_eval_map]
   exact H hp'
-#align is_integral_tower_bot_of_is_integral isIntegral_tower_bot_of_isIntegral
+#align is_integral_tower_bot_of_is_integral IsIntegral.tower_bot
 -/
 
-#print RingHom.isIntegral_tower_bot_of_isIntegral /-
-theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injective g)
-    (hfg : (g.comp f).IsIntegral) : f.IsIntegral := fun x =>
-  @isIntegral_tower_bot_of_isIntegral R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
+#print RingHom.IsIntegral.tower_bot /-
+theorem RingHom.IsIntegral.tower_bot (hg : Function.Injective g) (hfg : (g.comp f).IsIntegral) :
+    f.IsIntegral := fun x =>
+  @IsIntegral.tower_bot R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
     (@IsScalarTower.of_algebraMap_eq R S T _ _ _ f.toAlgebra g.toAlgebra (g.comp f).toAlgebra
       (RingHom.comp_apply g f))
     hg x (hfg (g x))
-#align ring_hom.is_integral_tower_bot_of_is_integral RingHom.isIntegral_tower_bot_of_isIntegral
+#align ring_hom.is_integral_tower_bot_of_is_integral RingHom.IsIntegral.tower_bot
 -/
 
-#print isIntegral_tower_bot_of_isIntegral_field /-
-theorem isIntegral_tower_bot_of_isIntegral_field {R A B : Type _} [CommRing R] [Field A]
-    [CommRing B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]
-    {x : A} (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
-  isIntegral_tower_bot_of_isIntegral (algebraMap A B).Injective h
-#align is_integral_tower_bot_of_is_integral_field isIntegral_tower_bot_of_isIntegral_field
+#print IsIntegral.tower_bot_of_field /-
+theorem IsIntegral.tower_bot_of_field {R A B : Type _} [CommRing R] [Field A] [CommRing B]
+    [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B] {x : A}
+    (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
+  IsIntegral.tower_bot (algebraMap A B).Injective h
+#align is_integral_tower_bot_of_is_integral_field IsIntegral.tower_bot_of_field
 -/
 
-#print RingHom.isIntegralElem_of_isIntegralElem_comp /-
-theorem RingHom.isIntegralElem_of_isIntegralElem_comp {x : T} (h : (g.comp f).IsIntegralElem x) :
+#print RingHom.isIntegralElem.of_comp /-
+theorem RingHom.isIntegralElem.of_comp {x : T} (h : (g.comp f).IsIntegralElem x) :
     g.IsIntegralElem x :=
   let ⟨p, ⟨hp, hp'⟩⟩ := h
   ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp' ⟩
-#align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem_of_isIntegralElem_comp
+#align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem.of_comp
 -/
 
-#print RingHom.isIntegral_tower_top_of_isIntegral /-
-theorem RingHom.isIntegral_tower_top_of_isIntegral (h : (g.comp f).IsIntegral) : g.IsIntegral :=
-  fun x => RingHom.isIntegralElem_of_isIntegralElem_comp f g (h x)
-#align ring_hom.is_integral_tower_top_of_is_integral RingHom.isIntegral_tower_top_of_isIntegral
+#print RingHom.IsIntegral.tower_top /-
+theorem RingHom.IsIntegral.tower_top (h : (g.comp f).IsIntegral) : g.IsIntegral := fun x =>
+  RingHom.isIntegralElem.of_comp f g (h x)
+#align ring_hom.is_integral_tower_top_of_is_integral RingHom.IsIntegral.tower_top
 -/
 
-#print isIntegral_tower_top_of_isIntegral /-
+/- warning: is_integral_tower_top_of_is_integral clashes with is_integral_of_is_scalar_tower -> IsIntegral.tower_top
+Case conversion may be inaccurate. Consider using '#align is_integral_tower_top_of_is_integral IsIntegral.tower_topₓ'. -/
+#print IsIntegral.tower_top /-
 /-- If `R → A → B` is an algebra tower,
 then if the entire tower is an integral extension so is `A → B`. -/
-theorem isIntegral_tower_top_of_isIntegral {x : B} (h : IsIntegral R x) : IsIntegral A x :=
+theorem IsIntegral.tower_top {x : B} (h : IsIntegral R x) : IsIntegral A x :=
   by
   rcases h with ⟨p, ⟨hp, hp'⟩⟩
   refine' ⟨p.map (algebraMap R A), ⟨hp.map (algebraMap R A), _⟩⟩
   rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map] at hp' 
   exact hp'
-#align is_integral_tower_top_of_is_integral isIntegral_tower_top_of_isIntegral
+#align is_integral_tower_top_of_is_integral IsIntegral.tower_top
 -/
 
-#print RingHom.isIntegral_quotient_of_isIntegral /-
-theorem RingHom.isIntegral_quotient_of_isIntegral {I : Ideal S} (hf : f.IsIntegral) :
+#print RingHom.IsIntegral.quotient /-
+theorem RingHom.IsIntegral.quotient {I : Ideal S} (hf : f.IsIntegral) :
     (Ideal.quotientMap I f le_rfl).IsIntegral :=
   by
   rintro ⟨x⟩
   obtain ⟨p, ⟨p_monic, hpx⟩⟩ := hf x
   refine' ⟨p.map (Ideal.Quotient.mk _), ⟨p_monic.map _, _⟩⟩
   simpa only [hom_eval₂, eval₂_map] using congr_arg (Ideal.Quotient.mk I) hpx
-#align ring_hom.is_integral_quotient_of_is_integral RingHom.isIntegral_quotient_of_isIntegral
+#align ring_hom.is_integral_quotient_of_is_integral RingHom.IsIntegral.quotient
 -/
 
-#print isIntegral_quotient_of_isIntegral /-
-theorem isIntegral_quotient_of_isIntegral {I : Ideal A} (hRA : Algebra.IsIntegral R A) :
+#print Algebra.IsIntegral.quotient /-
+theorem Algebra.IsIntegral.quotient {I : Ideal A} (hRA : Algebra.IsIntegral R A) :
     Algebra.IsIntegral (R ⧸ I.comap (algebraMap R A)) (A ⧸ I) :=
-  (algebraMap R A).isIntegral_quotient_of_isIntegral hRA
-#align is_integral_quotient_of_is_integral isIntegral_quotient_of_isIntegral
+  (algebraMap R A).quotient hRA
+#align is_integral_quotient_of_is_integral Algebra.IsIntegral.quotient
 -/
 
 #print isIntegral_quotientMap_iff /-
@@ -1286,8 +1285,8 @@ theorem isIntegral_quotientMap_iff {I : Ideal S} :
   by
   let g := Ideal.Quotient.mk (I.comap f)
   have := Ideal.quotientMap_comp_mk le_rfl
-  refine' ⟨fun h => _, fun h => RingHom.isIntegral_tower_top_of_isIntegral g _ (this ▸ h)⟩
-  refine' this ▸ RingHom.isIntegral_trans g (Ideal.quotientMap I f le_rfl) _ h
+  refine' ⟨fun h => _, fun h => RingHom.IsIntegral.tower_top g _ (this ▸ h)⟩
+  refine' this ▸ RingHom.IsIntegral.trans g (Ideal.quotientMap I f le_rfl) _ h
   exact RingHom.isIntegral_of_surjective g Ideal.Quotient.mk_surjective
 #align is_integral_quotient_map_iff isIntegral_quotientMap_iff
 -/
@@ -1342,7 +1341,7 @@ theorem isField_of_isIntegral_of_isField' {R S : Type _} [CommRing R] [CommRing
   refine' ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun x hx => _⟩
   let A := Algebra.adjoin R ({x} : Set S)
   haveI : IsNoetherian R A :=
-    isNoetherian_of_fg_of_noetherian A.to_submodule (FG_adjoin_singleton_of_integral x (H x))
+    isNoetherian_of_fg_of_noetherian A.to_submodule (IsIntegral.fg_adjoin_singleton x (H x))
   haveI : Module.Finite R A := Module.IsNoetherian.finite R A
   obtain ⟨y, hy⟩ :=
     LinearMap.surjective_of_injective

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

@@ -3,14 +3,14 @@ Copyright (c) 2019 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 -/
-import Mathbin.Data.Polynomial.Expand
-import Mathbin.LinearAlgebra.FiniteDimensional
-import Mathbin.LinearAlgebra.Matrix.Charpoly.LinearMap
-import Mathbin.RingTheory.Adjoin.Fg
-import Mathbin.RingTheory.FiniteType
-import Mathbin.RingTheory.Polynomial.ScaleRoots
-import Mathbin.RingTheory.Polynomial.Tower
-import Mathbin.RingTheory.TensorProduct
+import Data.Polynomial.Expand
+import LinearAlgebra.FiniteDimensional
+import LinearAlgebra.Matrix.Charpoly.LinearMap
+import RingTheory.Adjoin.Fg
+import RingTheory.FiniteType
+import RingTheory.Polynomial.ScaleRoots
+import RingTheory.Polynomial.Tower
+import RingTheory.TensorProduct
 
 #align_import ring_theory.integral_closure from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
 
@@ -952,7 +952,7 @@ end
 section IsIntegralClosure
 
 #print IsIntegralClosure /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
 /-- `is_integral_closure A R B` is the characteristic predicate stating `A` is
 the integral closure of `R` in `B`,
 i.e. that an element of `B` is integral over `R` iff it is an element of (the image of) `A`.

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

@@ -448,7 +448,7 @@ theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
 #align ring_hom.finite.to_is_integral RingHom.Finite.to_isIntegral
 -/
 
-alias RingHom.Finite.to_isIntegral ← RingHom.IsIntegral.of_finite
+alias RingHom.IsIntegral.of_finite := RingHom.Finite.to_isIntegral
 #align ring_hom.is_integral.of_finite RingHom.IsIntegral.of_finite
 
 #print RingHom.IsIntegral.to_finite /-
@@ -463,7 +463,7 @@ theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.
 #align ring_hom.is_integral.to_finite RingHom.IsIntegral.to_finite
 -/
 
-alias RingHom.IsIntegral.to_finite ← RingHom.Finite.of_isIntegral_of_finiteType
+alias RingHom.Finite.of_isIntegral_of_finiteType := RingHom.IsIntegral.to_finite
 #align ring_hom.finite.of_is_integral_of_finite_type RingHom.Finite.of_isIntegral_of_finiteType
 
 #print RingHom.finite_iff_isIntegral_and_finiteType /-

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

@@ -355,7 +355,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   let S₁ : Subring A :=
     { carrier := span S₀ (insert 1 ↑y : Set A)
       one_mem' := subset_span <| Or.inl rfl
-      mul_mem' := fun p q hp hq => this <| mul_mem_mul hp hq
+      hMul_mem' := fun p q hp hq => this <| mul_mem_mul hp hq
       zero_mem' := (span S₀ (insert 1 ↑y : Set A)).zero_mem
       add_mem' := fun _ _ => (span S₀ (insert 1 ↑y : Set A)).add_mem
       neg_mem' := fun _ => (span S₀ (insert 1 ↑y : Set A)).neg_mem }
@@ -411,7 +411,7 @@ theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R
   by
   let A' : Subalgebra R A :=
     { carrier := {x | ∀ n ∈ N, x • n ∈ N}
-      mul_mem' := fun a b ha hb n hn => smul_smul a b n ▸ ha _ (hb _ hn)
+      hMul_mem' := fun a b ha hb n hn => smul_smul a b n ▸ ha _ (hb _ hn)
       one_mem' := fun n hn => (one_smul A n).symm ▸ hn
       add_mem' := fun a b ha hb n hn => (add_smul a b n).symm ▸ N.add_mem (ha _ hn) (hb _ hn)
       zero_mem' := fun n hn => (zero_smul A n).symm ▸ N.zero_mem
@@ -630,7 +630,7 @@ def integralClosure : Subalgebra R A
   zero_mem' := isIntegral_zero
   one_mem' := isIntegral_one
   add_mem' _ _ := isIntegral_add
-  mul_mem' _ _ := isIntegral_mul
+  hMul_mem' _ _ := isIntegral_mul
   algebraMap_mem' x := isIntegral_algebraMap
 #align integral_closure integralClosure
 -/

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

@@ -810,7 +810,7 @@ theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x 
     simp only [AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom, mul_one, one_mul,
       Algebra.TensorProduct.includeLeft_apply, Algebra.TensorProduct.tmul_mul_tmul]
   convert (MulZeroClass.mul_zero _).symm
-  rw [Polynomial.eval₂_map, Algebra.TensorProduct.includeLeft_comp_algebraMap, ←
+  rw [Polynomial.eval₂_map, Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap, ←
     Polynomial.eval₂_map]
   convert Polynomial.eval₂_at_apply algebra.tensor_product.include_right.to_ring_hom y
   rw [Polynomial.eval_map, hp', _root_.map_zero]

mathlib commit https://github.com/leanprover-community/mathlib/commit/63721b2c3eba6c325ecf8ae8cca27155a4f6306f

@@ -1306,7 +1306,7 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
   -- and `q` be `p` with coefficients reversed (so `q(a) = q'(a) * a + 1`).
   -- We claim that `q(a) = 0`, so `-q'(a)` is the inverse of `a`.
   obtain ⟨p, p_monic, hp⟩ := H a_inv
-  use -∑ i : ℕ in Finset.range p.nat_degree, p.coeff i * a ^ (p.nat_degree - i - 1)
+  use-∑ i : ℕ in Finset.range p.nat_degree, p.coeff i * a ^ (p.nat_degree - i - 1)
   -- `q(a) = 0`, because multiplying everything with `a_inv^n` gives `p(a_inv) = 0`.
   -- TODO: this could be a lemma for `polynomial.reverse`.
   have hq : ∑ i : ℕ in Finset.range (p.nat_degree + 1), p.coeff i * a ^ (p.nat_degree - i) = 0 :=

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

@@ -2,11 +2,6 @@
 Copyright (c) 2019 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
-
-! This file was ported from Lean 3 source module ring_theory.integral_closure
-! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Data.Polynomial.Expand
 import Mathbin.LinearAlgebra.FiniteDimensional
@@ -17,6 +12,8 @@ import Mathbin.RingTheory.Polynomial.ScaleRoots
 import Mathbin.RingTheory.Polynomial.Tower
 import Mathbin.RingTheory.TensorProduct
 
+#align_import ring_theory.integral_closure from "leanprover-community/mathlib"@"38df578a6450a8c5142b3727e3ae894c2300cae0"
+
 /-!
 # Integral closure of a subring.
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/728ef9dbb281241906f25cbeb30f90d83e0bb451

@@ -701,21 +701,21 @@ theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
 #align integral_closure.is_integral integralClosure.isIntegral
 -/
 
-#print RingHom.is_integral_of_is_integral_mul_unit /-
-theorem RingHom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
+#print RingHom.isIntegral_of_isIntegral_mul_unit /-
+theorem RingHom.isIntegral_of_isIntegral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
     (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x :=
   by
   obtain ⟨p, ⟨p_monic, hp⟩⟩ := hx
   refine' ⟨scale_roots p r, ⟨(monic_scale_roots_iff r).2 p_monic, _⟩⟩
   convert scale_roots_eval₂_eq_zero f hp
   rw [mul_comm x y, ← mul_assoc, hr, one_mul]
-#align ring_hom.is_integral_of_is_integral_mul_unit RingHom.is_integral_of_is_integral_mul_unit
+#align ring_hom.is_integral_of_is_integral_mul_unit RingHom.isIntegral_of_isIntegral_mul_unit
 -/
 
 #print isIntegral_of_isIntegral_mul_unit /-
 theorem isIntegral_of_isIntegral_mul_unit {x y : A} {r : R} (hr : algebraMap R A r * y = 1)
     (hx : IsIntegral R (x * y)) : IsIntegral R x :=
-  (algebraMap R A).is_integral_of_is_integral_mul_unit x y r hr hx
+  (algebraMap R A).isIntegral_of_isIntegral_mul_unit x y r hr hx
 #align is_integral_of_is_integral_mul_unit isIntegral_of_isIntegral_mul_unit
 -/
 
@@ -728,11 +728,11 @@ theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x
 #align is_integral_of_mem_closure' isIntegral_of_mem_closure'
 -/
 
-#print is_integral_of_mem_closure'' /-
-theorem is_integral_of_mem_closure'' {S : Type _} [CommRing S] {f : R →+* S} (G : Set S)
+#print isIntegral_of_mem_closure'' /-
+theorem isIntegral_of_mem_closure'' {S : Type _} [CommRing S] {f : R →+* S} (G : Set S)
     (hG : ∀ x ∈ G, f.IsIntegralElem x) : ∀ x ∈ Subring.closure G, f.IsIntegralElem x := fun x hx =>
   @isIntegral_of_mem_closure' R S _ _ f.toAlgebra G hG x hx
-#align is_integral_of_mem_closure'' is_integral_of_mem_closure''
+#align is_integral_of_mem_closure'' isIntegral_of_mem_closure''
 -/
 
 #print IsIntegral.pow /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/8b981918a93bc45a8600de608cde7944a80d92b9

@@ -371,7 +371,9 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
     constructor <;> intro hz
     ·
       exact
-        (span_le.2 (Set.insert_subset.2 ⟨(Algebra.adjoin S₀ ↑y).one_mem, Algebra.subset_adjoin⟩)) hz
+        (span_le.2
+            (Set.insert_subset_iff.2 ⟨(Algebra.adjoin S₀ ↑y).one_mem, Algebra.subset_adjoin⟩))
+          hz
     · rw [Subalgebra.mem_toSubmodule, Algebra.mem_adjoin_iff] at hz 
       suffices Subring.closure (Set.range ⇑(algebraMap (↥S₀) A) ∪ ↑y) ≤ S₁ by exact this hz
       refine' Subring.closure_le.2 (Set.union_subset _ fun t ht => subset_span <| Or.inr ht)

mathlib commit https://github.com/leanprover-community/mathlib/commit/f2ad3645af9effcdb587637dc28a6074edc813f9

@@ -1098,8 +1098,8 @@ variable [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A'
 /-- Integral closures are all isomorphic to each other. -/
 noncomputable def equiv : A ≃ₐ[R] A' :=
   AlgEquiv.ofAlgHom (lift _ B (isIntegral_algebra R B)) (lift _ B (isIntegral_algebra R B))
-    (by ext x; apply algebra_map_injective A' R B; simp)
-    (by ext x; apply algebra_map_injective A R B; simp)
+    (by ext x; apply algebraMap_injective A' R B; simp)
+    (by ext x; apply algebraMap_injective A R B; simp)
 #align is_integral_closure.equiv IsIntegralClosure.equiv
 -/
 

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

@@ -89,14 +89,19 @@ protected def Algebra.IsIntegral : Prop :=
 
 variable {R A}
 
+#print RingHom.is_integral_map /-
 theorem RingHom.is_integral_map {x : R} : f.IsIntegralElem (f x) :=
   ⟨X - C x, monic_X_sub_C _, by simp⟩
 #align ring_hom.is_integral_map RingHom.is_integral_map
+-/
 
+#print isIntegral_algebraMap /-
 theorem isIntegral_algebraMap {x : R} : IsIntegral R (algebraMap R A x) :=
   (algebraMap R A).is_integral_map
 #align is_integral_algebra_map isIntegral_algebraMap
+-/
 
+#print isIntegral_of_noetherian /-
 theorem isIntegral_of_noetherian (H : IsNoetherian R A) (x : A) : IsIntegral R x :=
   by
   let leval : R[X] →ₗ[R] A := (aeval x).toLinearMap
@@ -116,7 +121,9 @@ theorem isIntegral_of_noetherian (H : IsNoetherian R A) (x : A) : IsIntegral R x
   show leval (X ^ (N + 1) - p) = 0
   rw [LinearMap.map_sub, hpe, sub_self]
 #align is_integral_of_noetherian isIntegral_of_noetherian
+-/
 
+#print isIntegral_of_submodule_noetherian /-
 theorem isIntegral_of_submodule_noetherian (S : Subalgebra R A) (H : IsNoetherian R S.toSubmodule)
     (x : A) (hx : x ∈ S) : IsIntegral R x :=
   by
@@ -131,6 +138,7 @@ theorem isIntegral_of_submodule_noetherian (S : Subalgebra R A) (H : IsNoetheria
     rw [S.val.map_mul, S.val.map_pow, S.val.commutes, S.val_apply, Subtype.coe_mk]
   refine' isIntegral_of_noetherian H ⟨x, hx⟩
 #align is_integral_of_submodule_noetherian isIntegral_of_submodule_noetherian
+-/
 
 end Ring
 
@@ -142,6 +150,7 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S]
 
 variable [Algebra R A] [Algebra R B] (f : R →+* S)
 
+#print map_isIntegral /-
 theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
     [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B} [AlgHomClass F A B C] (f : F)
     (hb : IsIntegral R b) : IsIntegral R (f b) :=
@@ -151,7 +160,9 @@ theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra
   rw [← aeval_def, show (aeval (f b)) P = (aeval (f b)) (P.map (algebraMap R A)) by simp,
     aeval_alg_hom_apply, aeval_map_algebra_map, aeval_def, hP.2, _root_.map_zero]
 #align map_is_integral map_isIntegral
+-/
 
+#print isIntegral_map_of_comp_eq_of_isIntegral /-
 theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R] [CommRing S]
     [CommRing T] [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
     (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
@@ -162,7 +173,9 @@ theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R]
   rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply, eval_map, hp.right,
     RingHom.map_zero]
 #align is_integral_map_of_comp_eq_of_is_integral isIntegral_map_of_comp_eq_of_isIntegral
+-/
 
+#print isIntegral_algHom_iff /-
 theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A →ₐ[R] B) (hf : Function.Injective f) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
   by
@@ -172,28 +185,38 @@ theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Al
   rwa [← f.comp_algebra_map, ← AlgHom.coe_toRingHom, ← Polynomial.hom_eval₂, AlgHom.coe_toRingHom,
     map_eq_zero_iff f hf] at hx 
 #align is_integral_alg_hom_iff isIntegral_algHom_iff
+-/
 
+#print isIntegral_algEquiv /-
 @[simp]
 theorem isIntegral_algEquiv {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
   ⟨fun h => by simpa using map_isIntegral f.symm.to_alg_hom h, map_isIntegral f.toAlgHom⟩
 #align is_integral_alg_equiv isIntegral_algEquiv
+-/
 
+#print isIntegral_of_isScalarTower /-
 theorem isIntegral_of_isScalarTower [Algebra A B] [IsScalarTower R A B] {x : B}
     (hx : IsIntegral R x) : IsIntegral A x :=
   let ⟨p, hp, hpx⟩ := hx
   ⟨p.map <| algebraMap R A, hp.map _, by rw [← aeval_def, aeval_map_algebra_map, aeval_def, hpx]⟩
 #align is_integral_of_is_scalar_tower isIntegral_of_isScalarTower
+-/
 
+#print map_isIntegral_int /-
 theorem map_isIntegral_int {B C F : Type _} [Ring B] [Ring C] {b : B} [RingHomClass F B C] (f : F)
     (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
   map_isIntegral (f : B →+* C).toIntAlgHom hb
 #align map_is_integral_int map_isIntegral_int
+-/
 
+#print isIntegral_ofSubring /-
 theorem isIntegral_ofSubring {x : A} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
   isIntegral_of_isScalarTower hx
 #align is_integral_of_subring isIntegral_ofSubring
+-/
 
+#print IsIntegral.algebraMap /-
 theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A} (h : IsIntegral R x) :
     IsIntegral R (algebraMap A B x) :=
   by
@@ -201,13 +224,17 @@ theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A} (h : I
   use f, hf
   rw [IsScalarTower.algebraMap_eq R A B, ← hom_eval₂, hx, RingHom.map_zero]
 #align is_integral.algebra_map IsIntegral.algebraMap
+-/
 
+#print isIntegral_algebraMap_iff /-
 theorem isIntegral_algebraMap_iff [Algebra A B] [IsScalarTower R A B] {x : A}
     (hAB : Function.Injective (algebraMap A B)) :
     IsIntegral R (algebraMap A B x) ↔ IsIntegral R x :=
   isIntegral_algHom_iff (IsScalarTower.toAlgHom R A B) hAB
 #align is_integral_algebra_map_iff isIntegral_algebraMap_iff
+-/
 
+#print isIntegral_iff_isIntegral_closure_finite /-
 theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
     IsIntegral R r ↔ ∃ s : Set R, s.Finite ∧ IsIntegral (Subring.closure s) r :=
   by
@@ -218,7 +245,9 @@ theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
   rcases hr with ⟨s, hs, hsr⟩
   exact isIntegral_ofSubring _ hsr
 #align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finite
+-/
 
+#print FG_adjoin_singleton_of_integral /-
 theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
     (Algebra.adjoin R ({x} : Set A)).toSubmodule.FG :=
   by
@@ -246,7 +275,9 @@ theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rw [mem_support_iff] at hkq ; apply hkq; apply this
   exact lt_of_le_of_lt degree_le_nat_degree (WithBot.coe_lt_coe.2 hk)
 #align fg_adjoin_singleton_of_integral FG_adjoin_singleton_of_integral
+-/
 
+#print FG_adjoin_of_finite /-
 theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
     (Algebra.adjoin R s).toSubmodule.FG :=
   Set.Finite.induction_on hfs
@@ -264,13 +295,17 @@ theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
             (FG_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s))
     his
 #align fg_adjoin_of_finite FG_adjoin_of_finite
+-/
 
+#print isNoetherian_adjoin_finset /-
 theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
     (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (↑s : Set A)) :=
   isNoetherian_of_fg_of_noetherian _ (FG_adjoin_of_finite s.finite_toSet hs)
 #align is_noetherian_adjoin_finset isNoetherian_adjoin_finset
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print isIntegral_of_mem_of_FG /-
 /-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
   then all elements of `S` are integral over `R`. -/
 theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) :
@@ -359,12 +394,16 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   rw [Finsupp.mem_supported] at hlx1 
   exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _
 #align is_integral_of_mem_of_fg isIntegral_of_mem_of_FG
+-/
 
+#print Module.End.isIntegral /-
 theorem Module.End.isIntegral {M : Type _} [AddCommGroup M] [Module R M] [Module.Finite R M] :
     Algebra.IsIntegral R (Module.End R M) :=
   LinearMap.exists_monic_and_aeval_eq_zero R
 #align module.End.is_integral Module.End.isIntegral
+-/
 
+#print isIntegral_of_smul_mem_submodule /-
 /-- Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
 and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. -/
 theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R M] [Module A M]
@@ -399,17 +438,21 @@ theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R
   haveI : Module.Finite R N := by rwa [Module.finite_def, Submodule.fg_top]
   apply Module.End.isIntegral
 #align is_integral_of_smul_mem_submodule isIntegral_of_smul_mem_submodule
+-/
 
 variable {f}
 
+#print RingHom.Finite.to_isIntegral /-
 theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
   letI := f.to_algebra
   fun x => isIntegral_of_mem_of_FG ⊤ h.1 _ trivial
 #align ring_hom.finite.to_is_integral RingHom.Finite.to_isIntegral
+-/
 
 alias RingHom.Finite.to_isIntegral ← RingHom.IsIntegral.of_finite
 #align ring_hom.is_integral.of_finite RingHom.IsIntegral.of_finite
 
+#print RingHom.IsIntegral.to_finite /-
 theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite :=
   by
   letI := f.to_algebra
@@ -419,15 +462,19 @@ theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.
   rw [← hs]
   exact FG_adjoin_of_finite (Set.toFinite _) fun x _ => h x
 #align ring_hom.is_integral.to_finite RingHom.IsIntegral.to_finite
+-/
 
 alias RingHom.IsIntegral.to_finite ← RingHom.Finite.of_isIntegral_of_finiteType
 #align ring_hom.finite.of_is_integral_of_finite_type RingHom.Finite.of_isIntegral_of_finiteType
 
+#print RingHom.finite_iff_isIntegral_and_finiteType /-
 /-- finite = integral + finite type -/
 theorem RingHom.finite_iff_isIntegral_and_finiteType : f.Finite ↔ f.IsIntegral ∧ f.FiniteType :=
   ⟨fun h => ⟨h.to_isIntegral, h.to_finiteType⟩, fun ⟨h, h'⟩ => h.toFinite h'⟩
 #align ring_hom.finite_iff_is_integral_and_finite_type RingHom.finite_iff_isIntegral_and_finiteType
+-/
 
+#print Algebra.IsIntegral.finite /-
 theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.FiniteType R A] :
     Module.Finite R A :=
   by
@@ -435,21 +482,27 @@ theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.Fin
     h.to_finite (by delta RingHom.FiniteType; convert h'; ext; exact (Algebra.smul_def _ _).symm)
   delta RingHom.Finite at this ; convert this; ext; exact Algebra.smul_def _ _
 #align algebra.is_integral.finite Algebra.IsIntegral.finite
+-/
 
+#print Algebra.IsIntegral.of_finite /-
 theorem Algebra.IsIntegral.of_finite [h : Module.Finite R A] : Algebra.IsIntegral R A :=
   by
   apply RingHom.Finite.to_isIntegral
   delta RingHom.Finite; convert h; ext; exact (Algebra.smul_def _ _).symm
 #align algebra.is_integral.of_finite Algebra.IsIntegral.of_finite
+-/
 
+#print Algebra.finite_iff_isIntegral_and_finiteType /-
 /-- finite = integral + finite type -/
 theorem Algebra.finite_iff_isIntegral_and_finiteType :
     Module.Finite R A ↔ Algebra.IsIntegral R A ∧ Algebra.FiniteType R A :=
   ⟨fun h => ⟨Algebra.IsIntegral.of_finite, inferInstance⟩, fun ⟨h, h'⟩ => h.finite⟩
 #align algebra.finite_iff_is_integral_and_finite_type Algebra.finite_iff_isIntegral_and_finiteType
+-/
 
 variable (f)
 
+#print RingHom.is_integral_of_mem_closure /-
 theorem RingHom.is_integral_of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
     (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z :=
   by
@@ -460,67 +513,94 @@ theorem RingHom.is_integral_of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
     isIntegral_of_mem_of_FG (Algebra.adjoin R {x, y}) this z
       (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)
 #align ring_hom.is_integral_of_mem_closure RingHom.is_integral_of_mem_closure
+-/
 
+#print isIntegral_of_mem_closure /-
 theorem isIntegral_of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
     (hz : z ∈ Subring.closure ({x, y} : Set A)) : IsIntegral R z :=
   (algebraMap R A).is_integral_of_mem_closure hx hy hz
 #align is_integral_of_mem_closure isIntegral_of_mem_closure
+-/
 
+#print RingHom.is_integral_zero /-
 theorem RingHom.is_integral_zero : f.IsIntegralElem 0 :=
   f.map_zero ▸ f.is_integral_map
 #align ring_hom.is_integral_zero RingHom.is_integral_zero
+-/
 
+#print isIntegral_zero /-
 theorem isIntegral_zero : IsIntegral R (0 : A) :=
   (algebraMap R A).is_integral_zero
 #align is_integral_zero isIntegral_zero
+-/
 
+#print RingHom.is_integral_one /-
 theorem RingHom.is_integral_one : f.IsIntegralElem 1 :=
   f.map_one ▸ f.is_integral_map
 #align ring_hom.is_integral_one RingHom.is_integral_one
+-/
 
+#print isIntegral_one /-
 theorem isIntegral_one : IsIntegral R (1 : A) :=
   (algebraMap R A).is_integral_one
 #align is_integral_one isIntegral_one
+-/
 
+#print RingHom.is_integral_add /-
 theorem RingHom.is_integral_add {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x + y) :=
   f.is_integral_of_mem_closure hx hy <|
     Subring.add_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl))
 #align ring_hom.is_integral_add RingHom.is_integral_add
+-/
 
+#print isIntegral_add /-
 theorem isIntegral_add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x + y) :=
   (algebraMap R A).is_integral_add hx hy
 #align is_integral_add isIntegral_add
+-/
 
+#print RingHom.is_integral_neg /-
 theorem RingHom.is_integral_neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) :=
   f.is_integral_of_mem_closure hx hx (Subring.neg_mem _ (Subring.subset_closure (Or.inl rfl)))
 #align ring_hom.is_integral_neg RingHom.is_integral_neg
+-/
 
+#print isIntegral_neg /-
 theorem isIntegral_neg {x : A} (hx : IsIntegral R x) : IsIntegral R (-x) :=
   (algebraMap R A).is_integral_neg hx
 #align is_integral_neg isIntegral_neg
+-/
 
+#print RingHom.is_integral_sub /-
 theorem RingHom.is_integral_sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x - y) := by
   simpa only [sub_eq_add_neg] using f.is_integral_add hx (f.is_integral_neg hy)
 #align ring_hom.is_integral_sub RingHom.is_integral_sub
+-/
 
+#print isIntegral_sub /-
 theorem isIntegral_sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x - y) :=
   (algebraMap R A).is_integral_sub hx hy
 #align is_integral_sub isIntegral_sub
+-/
 
+#print RingHom.is_integral_mul /-
 theorem RingHom.is_integral_mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x * y) :=
   f.is_integral_of_mem_closure hx hy
     (Subring.mul_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl)))
 #align ring_hom.is_integral_mul RingHom.is_integral_mul
+-/
 
+#print isIntegral_mul /-
 theorem isIntegral_mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x * y) :=
   (algebraMap R A).is_integral_mul hx hy
 #align is_integral_mul isIntegral_mul
+-/
 
 #print isIntegral_smul /-
 theorem isIntegral_smul [Algebra S A] [Algebra R S] [IsScalarTower R S A] {x : A} (r : R)
@@ -531,6 +611,7 @@ theorem isIntegral_smul [Algebra S A] [Algebra R S] [IsScalarTower R S A] {x : A
 #align is_integral_smul isIntegral_smul
 -/
 
+#print isIntegral_of_pow /-
 theorem isIntegral_of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
     IsIntegral R x := by
   rcases hx with ⟨p, ⟨hmonic, heval⟩⟩
@@ -538,6 +619,7 @@ theorem isIntegral_of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x
     ⟨expand R n p, monic.expand hn hmonic, by
       rwa [eval₂_eq_eval_map, map_expand, expand_eval, ← eval₂_eq_eval_map]⟩
 #align is_integral_of_pow isIntegral_of_pow
+-/
 
 variable (R A)
 
@@ -554,15 +636,18 @@ def integralClosure : Subalgebra R A
 #align integral_closure integralClosure
 -/
 
+#print mem_integralClosure_iff_mem_FG /-
 theorem mem_integralClosure_iff_mem_FG {r : A} :
     r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M :=
   ⟨fun hr =>
     ⟨Algebra.adjoin R {r}, FG_adjoin_singleton_of_integral _ hr, Algebra.subset_adjoin rfl⟩,
     fun ⟨M, Hf, hrM⟩ => isIntegral_of_mem_of_FG M Hf _ hrM⟩
 #align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_FG
+-/
 
 variable {R} {A}
 
+#print adjoin_le_integralClosure /-
 theorem adjoin_le_integralClosure {x : A} (hx : IsIntegral R x) :
     Algebra.adjoin R {x} ≤ integralClosure R A :=
   by
@@ -570,7 +655,9 @@ theorem adjoin_le_integralClosure {x : A} (hx : IsIntegral R x) :
   simp only [SetLike.mem_coe, Set.singleton_subset_iff]
   exact hx
 #align adjoin_le_integral_closure adjoin_le_integralClosure
+-/
 
+#print le_integralClosure_iff_isIntegral /-
 theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
     S ≤ integralClosure R A ↔ Algebra.IsIntegral R S :=
   SetLike.forall.symm.trans
@@ -578,12 +665,16 @@ theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
       show IsIntegral R (algebraMap S A x) ↔ IsIntegral R x from
         isIntegral_algebraMap_iff Subtype.coe_injective)
 #align le_integral_closure_iff_is_integral le_integralClosure_iff_isIntegral
+-/
 
+#print isIntegral_sup /-
 theorem isIntegral_sup {S T : Subalgebra R A} :
     Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R S ∧ Algebra.IsIntegral R T := by
   simp only [← le_integralClosure_iff_isIntegral, sup_le_iff]
 #align is_integral_sup isIntegral_sup
+-/
 
+#print integralClosure_map_algEquiv /-
 /-- Mapping an integral closure along an `alg_equiv` gives the integral closure. -/
 theorem integralClosure_map_algEquiv (f : A ≃ₐ[R] B) :
     (integralClosure R A).map (f : A →ₐ[R] B) = integralClosure R B :=
@@ -597,14 +688,18 @@ theorem integralClosure_map_algEquiv (f : A ≃ₐ[R] B) :
     use f.symm y, map_isIntegral (f.symm : B →ₐ[R] A) hy
     simp
 #align integral_closure_map_alg_equiv integralClosure_map_algEquiv
+-/
 
+#print integralClosure.isIntegral /-
 theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
   let ⟨p, hpm, hpx⟩ := x.2
   ⟨p, hpm,
     Subtype.eq <| by
       rwa [← aeval_def, Subtype.val_eq_coe, ← Subalgebra.val_apply, aeval_alg_hom_apply] at hpx ⟩
 #align integral_closure.is_integral integralClosure.isIntegral
+-/
 
+#print RingHom.is_integral_of_is_integral_mul_unit /-
 theorem RingHom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
     (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x :=
   by
@@ -613,35 +708,48 @@ theorem RingHom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r
   convert scale_roots_eval₂_eq_zero f hp
   rw [mul_comm x y, ← mul_assoc, hr, one_mul]
 #align ring_hom.is_integral_of_is_integral_mul_unit RingHom.is_integral_of_is_integral_mul_unit
+-/
 
+#print isIntegral_of_isIntegral_mul_unit /-
 theorem isIntegral_of_isIntegral_mul_unit {x y : A} {r : R} (hr : algebraMap R A r * y = 1)
     (hx : IsIntegral R (x * y)) : IsIntegral R x :=
   (algebraMap R A).is_integral_of_is_integral_mul_unit x y r hr hx
 #align is_integral_of_is_integral_mul_unit isIntegral_of_isIntegral_mul_unit
+-/
 
+#print isIntegral_of_mem_closure' /-
 /-- Generalization of `is_integral_of_mem_closure` bootstrapped up from that lemma -/
 theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) :
     ∀ x ∈ Subring.closure G, IsIntegral R x := fun x hx =>
   Subring.closure_induction hx hG isIntegral_zero isIntegral_one (fun _ _ => isIntegral_add)
     (fun _ => isIntegral_neg) fun _ _ => isIntegral_mul
 #align is_integral_of_mem_closure' isIntegral_of_mem_closure'
+-/
 
+#print is_integral_of_mem_closure'' /-
 theorem is_integral_of_mem_closure'' {S : Type _} [CommRing S] {f : R →+* S} (G : Set S)
     (hG : ∀ x ∈ G, f.IsIntegralElem x) : ∀ x ∈ Subring.closure G, f.IsIntegralElem x := fun x hx =>
   @isIntegral_of_mem_closure' R S _ _ f.toAlgebra G hG x hx
 #align is_integral_of_mem_closure'' is_integral_of_mem_closure''
+-/
 
+#print IsIntegral.pow /-
 theorem IsIntegral.pow {x : A} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (x ^ n) :=
   (integralClosure R A).pow_mem h n
 #align is_integral.pow IsIntegral.pow
+-/
 
+#print IsIntegral.nsmul /-
 theorem IsIntegral.nsmul {x : A} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (n • x) :=
   (integralClosure R A).nsmul_mem h n
 #align is_integral.nsmul IsIntegral.nsmul
+-/
 
+#print IsIntegral.zsmul /-
 theorem IsIntegral.zsmul {x : A} (h : IsIntegral R x) (n : ℤ) : IsIntegral R (n • x) :=
   (integralClosure R A).zsmul_mem h n
 #align is_integral.zsmul IsIntegral.zsmul
+-/
 
 #print IsIntegral.multiset_prod /-
 theorem IsIntegral.multiset_prod {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) :
@@ -657,15 +765,19 @@ theorem IsIntegral.multiset_sum {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R
 #align is_integral.multiset_sum IsIntegral.multiset_sum
 -/
 
+#print IsIntegral.prod /-
 theorem IsIntegral.prod {α : Type _} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
     IsIntegral R (∏ x in s, f x) :=
   (integralClosure R A).prod_mem h
 #align is_integral.prod IsIntegral.prod
+-/
 
+#print IsIntegral.sum /-
 theorem IsIntegral.sum {α : Type _} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
     IsIntegral R (∑ x in s, f x) :=
   (integralClosure R A).sum_mem h
 #align is_integral.sum IsIntegral.sum
+-/
 
 #print IsIntegral.det /-
 theorem IsIntegral.det {n : Type _} [Fintype n] [DecidableEq n] {M : Matrix n n A}
@@ -676,13 +788,16 @@ theorem IsIntegral.det {n : Type _} [Fintype n] [DecidableEq n] {M : Matrix n n
 #align is_integral.det IsIntegral.det
 -/
 
+#print IsIntegral.pow_iff /-
 @[simp]
 theorem IsIntegral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n) ↔ IsIntegral R x :=
   ⟨isIntegral_of_pow hn, fun hx => IsIntegral.pow hx n⟩
 #align is_integral.pow_iff IsIntegral.pow_iff
+-/
 
 open scoped TensorProduct
 
+#print IsIntegral.tmul /-
 theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x ⊗ₜ[R] y) :=
   by
   obtain ⟨p, hp, hp'⟩ := h
@@ -701,6 +816,7 @@ theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x 
   convert Polynomial.eval₂_at_apply algebra.tensor_product.include_right.to_ring_hom y
   rw [Polynomial.eval_map, hp', _root_.map_zero]
 #align is_integral.tmul IsIntegral.tmul
+-/
 
 section
 
@@ -714,6 +830,7 @@ noncomputable def normalizeScaleRoots (p : R[X]) : R[X] :=
 #align normalize_scale_roots normalizeScaleRoots
 -/
 
+#print normalizeScaleRoots_coeff_mul_leadingCoeff_pow /-
 theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ natDegree p) :
     (normalizeScaleRoots p).coeff i * p.leadingCoeff ^ i =
       p.coeff i * p.leadingCoeff ^ (p.natDegree - 1) :=
@@ -728,6 +845,7 @@ theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ nat
     rw [lt_iff_le_and_ne]
     exact ⟨le_nat_degree_of_ne_zero h₁, h₂⟩
 #align normalize_scale_roots_coeff_mul_leading_coeff_pow normalizeScaleRoots_coeff_mul_leadingCoeff_pow
+-/
 
 #print leadingCoeff_smul_normalizeScaleRoots /-
 theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
@@ -768,6 +886,7 @@ theorem normalizeScaleRoots_degree : (normalizeScaleRoots p).degree = p.degree :
 #align normalize_scale_roots_degree normalizeScaleRoots_degree
 -/
 
+#print normalizeScaleRoots_eval₂_leadingCoeff_mul /-
 theorem normalizeScaleRoots_eval₂_leadingCoeff_mul (h : 1 ≤ p.natDegree) (f : R →+* S) (x : S) :
     (normalizeScaleRoots p).eval₂ f (f p.leadingCoeff * x) =
       f p.leadingCoeff ^ (p.natDegree - 1) * p.eval₂ f x :=
@@ -780,6 +899,7 @@ theorem normalizeScaleRoots_eval₂_leadingCoeff_mul (h : 1 ≤ p.natDegree) (f
     normalizeScaleRoots_coeff_mul_leadingCoeff_pow _ _ h, f.map_mul, f.map_pow]
   ring
 #align normalize_scale_roots_eval₂_leading_coeff_mul normalizeScaleRoots_eval₂_leadingCoeff_mul
+-/
 
 #print normalizeScaleRoots_monic /-
 theorem normalizeScaleRoots_monic (h : p ≠ 0) : (normalizeScaleRoots p).Monic :=
@@ -791,6 +911,7 @@ theorem normalizeScaleRoots_monic (h : p ≠ 0) : (normalizeScaleRoots p).Monic
 #align normalize_scale_roots_monic normalizeScaleRoots_monic
 -/
 
+#print RingHom.isIntegralElem_leadingCoeff_mul /-
 /-- Given a `p : R[X]` and a `x : S` such that `p.eval₂ f x = 0`,
 `f p.leading_coeff * x` is integral. -/
 theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
@@ -811,7 +932,9 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
       suffices p.map f = 0 by exact (hp this).rec _
       rw [eq_C_of_nat_degree_eq_zero h', map_C, h, C_eq_zero]
 #align ring_hom.is_integral_elem_leading_coeff_mul RingHom.isIntegralElem_leadingCoeff_mul
+-/
 
+#print isIntegral_leadingCoeff_smul /-
 /-- Given a `p : R[X]` and a root `x : S`,
 then `p.leading_coeff • x : S` is integral over `R`. -/
 theorem isIntegral_leadingCoeff_smul [Algebra R S] (h : aeval x p = 0) :
@@ -821,6 +944,7 @@ theorem isIntegral_leadingCoeff_smul [Algebra R S] (h : aeval x p = 0) :
   rw [Algebra.smul_def]
   exact (algebraMap R S).isIntegralElem_leadingCoeff_mul p x h
 #align is_integral_leading_coeff_smul isIntegral_leadingCoeff_smul
+-/
 
 end
 
@@ -829,7 +953,7 @@ end
 section IsIntegralClosure
 
 #print IsIntegralClosure /-
-/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
 /-- `is_integral_closure A R B` is the characteristic predicate stating `A` is
 the integral closure of `R` in `B`,
 i.e. that an element of `B` is integral over `R` iff it is an element of (the image of) `A`.
@@ -841,10 +965,12 @@ class IsIntegralClosure (A R B : Type _) [CommRing R] [CommSemiring A] [CommRing
 #align is_integral_closure IsIntegralClosure
 -/
 
+#print integralClosure.isIntegralClosure /-
 instance integralClosure.isIntegralClosure (R A : Type _) [CommRing R] [CommRing A] [Algebra R A] :
     IsIntegralClosure (integralClosure R A) R A :=
   ⟨Subtype.coe_injective, fun x => ⟨fun h => ⟨⟨x, h⟩, rfl⟩, by rintro ⟨⟨_, h⟩, rfl⟩; exact h⟩⟩
 #align integral_closure.is_integral_closure integralClosure.isIntegralClosure
+-/
 
 namespace IsIntegralClosure
 
@@ -854,15 +980,20 @@ variable [Algebra R B] [Algebra A B] [IsIntegralClosure A R B]
 
 variable (R) {A} (B)
 
+#print IsIntegralClosure.isIntegral /-
 protected theorem isIntegral [Algebra R A] [IsScalarTower R A B] (x : A) : IsIntegral R x :=
   (isIntegral_algebraMap_iff (algebraMap_injective A R B)).mp <|
     show IsIntegral R (algebraMap A B x) from isIntegral_iff.mpr ⟨x, rfl⟩
 #align is_integral_closure.is_integral IsIntegralClosure.isIntegral
+-/
 
+#print IsIntegralClosure.isIntegral_algebra /-
 theorem isIntegral_algebra [Algebra R A] [IsScalarTower R A B] : Algebra.IsIntegral R A := fun x =>
   IsIntegralClosure.isIntegral R B x
 #align is_integral_closure.is_integral_algebra IsIntegralClosure.isIntegral_algebra
+-/
 
+#print IsIntegralClosure.noZeroSMulDivisors /-
 theorem noZeroSMulDivisors [Algebra R A] [IsScalarTower R A B] [NoZeroSMulDivisors R B] :
     NoZeroSMulDivisors R A :=
   by
@@ -871,6 +1002,7 @@ theorem noZeroSMulDivisors [Algebra R A] [IsScalarTower R A B] [NoZeroSMulDiviso
       (map_zero _) fun _ _ => _
   simp only [Algebra.algebraMap_eq_smul_one, IsScalarTower.smul_assoc]
 #align is_integral_closure.no_zero_smul_divisors IsIntegralClosure.noZeroSMulDivisors
+-/
 
 variable {R} (A) {B}
 
@@ -881,39 +1013,51 @@ noncomputable def mk' (x : B) (hx : IsIntegral R x) : A :=
 #align is_integral_closure.mk' IsIntegralClosure.mk'
 -/
 
+#print IsIntegralClosure.algebraMap_mk' /-
 @[simp]
 theorem algebraMap_mk' (x : B) (hx : IsIntegral R x) : algebraMap A B (mk' A x hx) = x :=
   Classical.choose_spec (isIntegral_iff.mp hx)
 #align is_integral_closure.algebra_map_mk' IsIntegralClosure.algebraMap_mk'
+-/
 
+#print IsIntegralClosure.mk'_one /-
 @[simp]
 theorem mk'_one (h : IsIntegral R (1 : B) := isIntegral_one) : mk' A 1 h = 1 :=
   algebraMap_injective A R B <| by rw [algebra_map_mk', RingHom.map_one]
 #align is_integral_closure.mk'_one IsIntegralClosure.mk'_one
+-/
 
+#print IsIntegralClosure.mk'_zero /-
 @[simp]
 theorem mk'_zero (h : IsIntegral R (0 : B) := isIntegral_zero) : mk' A 0 h = 0 :=
   algebraMap_injective A R B <| by rw [algebra_map_mk', RingHom.map_zero]
 #align is_integral_closure.mk'_zero IsIntegralClosure.mk'_zero
+-/
 
+#print IsIntegralClosure.mk'_add /-
 @[simp]
 theorem mk'_add (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
     mk' A (x + y) (isIntegral_add hx hy) = mk' A x hx + mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebra_map_mk', RingHom.map_add]
 #align is_integral_closure.mk'_add IsIntegralClosure.mk'_add
+-/
 
+#print IsIntegralClosure.mk'_mul /-
 @[simp]
 theorem mk'_mul (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
     mk' A (x * y) (isIntegral_mul hx hy) = mk' A x hx * mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebra_map_mk', RingHom.map_mul]
 #align is_integral_closure.mk'_mul IsIntegralClosure.mk'_mul
+-/
 
+#print IsIntegralClosure.mk'_algebraMap /-
 @[simp]
 theorem mk'_algebraMap [Algebra R A] [IsScalarTower R A B] (x : R)
     (h : IsIntegral R (algebraMap R B x) := isIntegral_algebraMap) :
     IsIntegralClosure.mk' A (algebraMap R B x) h = algebraMap R A x :=
   algebraMap_injective A R B <| by rw [algebra_map_mk', ← IsScalarTower.algebraMap_apply]
 #align is_integral_closure.mk'_algebra_map IsIntegralClosure.mk'_algebraMap
+-/
 
 section lift
 
@@ -935,10 +1079,12 @@ noncomputable def lift : S →ₐ[R] A
 #align is_integral_closure.lift IsIntegralClosure.lift
 -/
 
+#print IsIntegralClosure.algebraMap_lift /-
 @[simp]
 theorem algebraMap_lift (x : S) : algebraMap A B (lift A B h x) = algebraMap S B x :=
   algebraMap_mk' _ _ _
 #align is_integral_closure.algebra_map_lift IsIntegralClosure.algebraMap_lift
+-/
 
 end lift
 
@@ -957,10 +1103,12 @@ noncomputable def equiv : A ≃ₐ[R] A' :=
 #align is_integral_closure.equiv IsIntegralClosure.equiv
 -/
 
+#print IsIntegralClosure.algebraMap_equiv /-
 @[simp]
 theorem algebraMap_equiv (x : A) : algebraMap A' B (equiv R A B A' x) = algebraMap A B x :=
   algebraMap_lift _ _ _ _
 #align is_integral_closure.algebra_map_equiv IsIntegralClosure.algebraMap_equiv
+-/
 
 end Equiv
 
@@ -978,6 +1126,7 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S] [CommRing T]
 
 variable [Algebra A B] [Algebra R B] (f : R →+* S) (g : S →+* T)
 
+#print isIntegral_trans_aux /-
 theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x p = 0) :
     IsIntegral (adjoin R (↑(p.map <| algebraMap A B).frange : Set B)) x :=
   by
@@ -1001,9 +1150,11 @@ theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x
     replace hq := congr_arg (eval x) hq
     convert hq using 1 <;> symm <;> apply eval_map
 #align is_integral_trans_aux isIntegral_trans_aux
+-/
 
 variable [Algebra R A] [IsScalarTower R A B]
 
+#print isIntegral_trans /-
 /-- If A is an R-algebra all of whose elements are integral over R,
 and x is an element of an A-algebra that is integral over A, then x is integral over R.-/
 theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegral A x) :
@@ -1019,30 +1170,40 @@ theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegr
   · apply FG_adjoin_singleton_of_integral
     exact isIntegral_trans_aux _ pmonic hp
 #align is_integral_trans isIntegral_trans
+-/
 
+#print Algebra.isIntegral_trans /-
 /-- If A is an R-algebra all of whose elements are integral over R,
 and B is an A-algebra all of whose elements are integral over A,
 then all elements of B are integral over R.-/
 theorem Algebra.isIntegral_trans (hA : Algebra.IsIntegral R A) (hB : Algebra.IsIntegral A B) :
     Algebra.IsIntegral R B := fun x => isIntegral_trans hA x (hB x)
 #align algebra.is_integral_trans Algebra.isIntegral_trans
+-/
 
+#print RingHom.isIntegral_trans /-
 theorem RingHom.isIntegral_trans (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.comp f).IsIntegral :=
   @Algebra.isIntegral_trans R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
     (@IsScalarTower.of_algebraMap_eq R S T _ _ _ f.toAlgebra g.toAlgebra (g.comp f).toAlgebra
       (RingHom.comp_apply g f))
     hf hg
 #align ring_hom.is_integral_trans RingHom.isIntegral_trans
+-/
 
+#print RingHom.isIntegral_of_surjective /-
 theorem RingHom.isIntegral_of_surjective (hf : Function.Surjective f) : f.IsIntegral := fun x =>
   (hf x).recOn fun y hy => (hy ▸ f.is_integral_map : f.IsIntegralElem x)
 #align ring_hom.is_integral_of_surjective RingHom.isIntegral_of_surjective
+-/
 
+#print isIntegral_of_surjective /-
 theorem isIntegral_of_surjective (h : Function.Surjective (algebraMap R A)) :
     Algebra.IsIntegral R A :=
   (algebraMap R A).isIntegral_of_surjective h
 #align is_integral_of_surjective isIntegral_of_surjective
+-/
 
+#print isIntegral_tower_bot_of_isIntegral /-
 /-- If `R → A → B` is an algebra tower with `A → B` injective,
 then if the entire tower is an integral extension so is `R → A` -/
 theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A B)) {x : A}
@@ -1055,7 +1216,9 @@ theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A
   rw [eval₂_eq_eval_map]
   exact H hp'
 #align is_integral_tower_bot_of_is_integral isIntegral_tower_bot_of_isIntegral
+-/
 
+#print RingHom.isIntegral_tower_bot_of_isIntegral /-
 theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injective g)
     (hfg : (g.comp f).IsIntegral) : f.IsIntegral := fun x =>
   @isIntegral_tower_bot_of_isIntegral R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
@@ -1063,23 +1226,31 @@ theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injective g)
       (RingHom.comp_apply g f))
     hg x (hfg (g x))
 #align ring_hom.is_integral_tower_bot_of_is_integral RingHom.isIntegral_tower_bot_of_isIntegral
+-/
 
+#print isIntegral_tower_bot_of_isIntegral_field /-
 theorem isIntegral_tower_bot_of_isIntegral_field {R A B : Type _} [CommRing R] [Field A]
     [CommRing B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]
     {x : A} (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
   isIntegral_tower_bot_of_isIntegral (algebraMap A B).Injective h
 #align is_integral_tower_bot_of_is_integral_field isIntegral_tower_bot_of_isIntegral_field
+-/
 
+#print RingHom.isIntegralElem_of_isIntegralElem_comp /-
 theorem RingHom.isIntegralElem_of_isIntegralElem_comp {x : T} (h : (g.comp f).IsIntegralElem x) :
     g.IsIntegralElem x :=
   let ⟨p, ⟨hp, hp'⟩⟩ := h
   ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp' ⟩
 #align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem_of_isIntegralElem_comp
+-/
 
+#print RingHom.isIntegral_tower_top_of_isIntegral /-
 theorem RingHom.isIntegral_tower_top_of_isIntegral (h : (g.comp f).IsIntegral) : g.IsIntegral :=
   fun x => RingHom.isIntegralElem_of_isIntegralElem_comp f g (h x)
 #align ring_hom.is_integral_tower_top_of_is_integral RingHom.isIntegral_tower_top_of_isIntegral
+-/
 
+#print isIntegral_tower_top_of_isIntegral /-
 /-- If `R → A → B` is an algebra tower,
 then if the entire tower is an integral extension so is `A → B`. -/
 theorem isIntegral_tower_top_of_isIntegral {x : B} (h : IsIntegral R x) : IsIntegral A x :=
@@ -1089,7 +1260,9 @@ theorem isIntegral_tower_top_of_isIntegral {x : B} (h : IsIntegral R x) : IsInte
   rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map] at hp' 
   exact hp'
 #align is_integral_tower_top_of_is_integral isIntegral_tower_top_of_isIntegral
+-/
 
+#print RingHom.isIntegral_quotient_of_isIntegral /-
 theorem RingHom.isIntegral_quotient_of_isIntegral {I : Ideal S} (hf : f.IsIntegral) :
     (Ideal.quotientMap I f le_rfl).IsIntegral :=
   by
@@ -1098,12 +1271,16 @@ theorem RingHom.isIntegral_quotient_of_isIntegral {I : Ideal S} (hf : f.IsIntegr
   refine' ⟨p.map (Ideal.Quotient.mk _), ⟨p_monic.map _, _⟩⟩
   simpa only [hom_eval₂, eval₂_map] using congr_arg (Ideal.Quotient.mk I) hpx
 #align ring_hom.is_integral_quotient_of_is_integral RingHom.isIntegral_quotient_of_isIntegral
+-/
 
+#print isIntegral_quotient_of_isIntegral /-
 theorem isIntegral_quotient_of_isIntegral {I : Ideal A} (hRA : Algebra.IsIntegral R A) :
     Algebra.IsIntegral (R ⧸ I.comap (algebraMap R A)) (A ⧸ I) :=
   (algebraMap R A).isIntegral_quotient_of_isIntegral hRA
 #align is_integral_quotient_of_is_integral isIntegral_quotient_of_isIntegral
+-/
 
+#print isIntegral_quotientMap_iff /-
 theorem isIntegral_quotientMap_iff {I : Ideal S} :
     (Ideal.quotientMap I f le_rfl).IsIntegral ↔
       ((Ideal.Quotient.mk I).comp f : R →+* S ⧸ I).IsIntegral :=
@@ -1114,7 +1291,9 @@ theorem isIntegral_quotientMap_iff {I : Ideal S} :
   refine' this ▸ RingHom.isIntegral_trans g (Ideal.quotientMap I f le_rfl) _ h
   exact RingHom.isIntegral_of_surjective g Ideal.Quotient.mk_surjective
 #align is_integral_quotient_map_iff isIntegral_quotientMap_iff
+-/
 
+#print isField_of_isIntegral_of_isField /-
 /-- If the integral extension `R → S` is injective, and `S` is a field, then `R` is also a field. -/
 theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial R] [CommRing S]
     [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S)
@@ -1154,7 +1333,9 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
   have : 1 ≤ p.nat_degree - i := le_tsub_of_add_le_left (finset.mem_range.mp hi)
   rw [mul_assoc, ← pow_succ', tsub_add_cancel_of_le this]
 #align is_field_of_is_integral_of_is_field isField_of_isIntegral_of_isField
+-/
 
+#print isField_of_isIntegral_of_isField' /-
 theorem isField_of_isIntegral_of_isField' {R S : Type _} [CommRing R] [CommRing S] [IsDomain S]
     [Algebra R S] (H : Algebra.IsIntegral R S) (hR : IsField R) : IsField S :=
   by
@@ -1171,15 +1352,19 @@ theorem isField_of_isIntegral_of_isField' {R S : Type _} [CommRing R] [CommRing
       1
   exact ⟨y, subtype.ext_iff.mp hy⟩
 #align is_field_of_is_integral_of_is_field' isField_of_isIntegral_of_isField'
+-/
 
+#print Algebra.IsIntegral.isField_iff_isField /-
 theorem Algebra.IsIntegral.isField_iff_isField {R S : Type _} [CommRing R] [Nontrivial R]
     [CommRing S] [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S)
     (hRS : Function.Injective (algebraMap R S)) : IsField R ↔ IsField S :=
   ⟨isField_of_isIntegral_of_isField' H, isField_of_isIntegral_of_isField H hRS⟩
 #align algebra.is_integral.is_field_iff_is_field Algebra.IsIntegral.isField_iff_isField
+-/
 
 end Algebra
 
+#print integralClosure_idem /-
 theorem integralClosure_idem {R : Type _} {A : Type _} [CommRing R] [CommRing A] [Algebra R A] :
     integralClosure (integralClosure R A : Set A) A = ⊥ :=
   eq_bot_iff.2 fun x hx =>
@@ -1189,6 +1374,7 @@ theorem integralClosure_idem {R : Type _} {A : Type _} [CommRing R] [CommRing A]
             integralClosure.isIntegral x hx⟩,
         rfl⟩
 #align integral_closure_idem integralClosure_idem
+-/
 
 section IsDomain
 
@@ -1197,10 +1383,12 @@ variable {R S : Type _} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S]
 instance : IsDomain (integralClosure R S) :=
   inferInstance
 
+#print roots_mem_integralClosure /-
 theorem roots_mem_integralClosure {f : R[X]} (hf : f.Monic) {a : S}
     (ha : a ∈ (f.map <| algebraMap R S).roots) : a ∈ integralClosure R S :=
   ⟨f, hf, (eval₂_eq_eval_map _).trans <| (mem_roots <| (hf.map _).NeZero).1 ha⟩
 #align roots_mem_integral_closure roots_mem_integralClosure
+-/
 
 end IsDomain
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -1131,7 +1131,7 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
   use -∑ i : ℕ in Finset.range p.nat_degree, p.coeff i * a ^ (p.nat_degree - i - 1)
   -- `q(a) = 0`, because multiplying everything with `a_inv^n` gives `p(a_inv) = 0`.
   -- TODO: this could be a lemma for `polynomial.reverse`.
-  have hq : (∑ i : ℕ in Finset.range (p.nat_degree + 1), p.coeff i * a ^ (p.nat_degree - i)) = 0 :=
+  have hq : ∑ i : ℕ in Finset.range (p.nat_degree + 1), p.coeff i * a ^ (p.nat_degree - i) = 0 :=
     by
     apply (injective_iff_map_eq_zero (algebraMap R S)).mp hRS
     have a_inv_ne_zero : a_inv ≠ 0 := right_ne_zero_of_mul (mt ha_inv.symm.trans one_ne_zero)

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

@@ -372,7 +372,7 @@ theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R
     (x : A) (hx : ∀ n ∈ N, x • n ∈ N) : IsIntegral R x :=
   by
   let A' : Subalgebra R A :=
-    { carrier := { x | ∀ n ∈ N, x • n ∈ N }
+    { carrier := {x | ∀ n ∈ N, x • n ∈ N}
       mul_mem' := fun a b ha hb n hn => smul_smul a b n ▸ ha _ (hb _ hn)
       one_mem' := fun n hn => (one_smul A n).symm ▸ hn
       add_mem' := fun a b ha hb n hn => (add_smul a b n).symm ▸ N.add_mem (ha _ hn) (hb _ hn)
@@ -385,7 +385,7 @@ theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R
         map_smul' := fun r s => LinearMap.ext fun n => Subtype.ext <| smul_assoc r s n }
       (LinearMap.ext fun n => Subtype.ext <| one_smul _ _) fun x y =>
       LinearMap.ext fun n => Subtype.ext <| mul_smul x y n
-  obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by by_contra h'; push_neg  at h' ; apply hN;
+  obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by by_contra h'; push_neg at h' ; apply hN;
     rwa [eq_bot_iff]
   have : Function.Injective f :=
     by
@@ -545,7 +545,7 @@ variable (R A)
 /-- The integral closure of R in an R-algebra A. -/
 def integralClosure : Subalgebra R A
     where
-  carrier := { r | IsIntegral R r }
+  carrier := {r | IsIntegral R r}
   zero_mem' := isIntegral_zero
   one_mem' := isIntegral_one
   add_mem' _ _ := isIntegral_add
@@ -688,13 +688,14 @@ theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x 
   obtain ⟨p, hp, hp'⟩ := h
   refine' ⟨(p.map (algebraMap R A)).scaleRoots x, _, _⟩
   · rw [Polynomial.monic_scaleRoots_iff]; exact hp.map _
-  convert@Polynomial.scaleRoots_eval₂_mul (A ⊗[R] B) A _ _ _
+  convert
+    @Polynomial.scaleRoots_eval₂_mul (A ⊗[R] B) A _ _ _
       algebra.tensor_product.include_left.to_ring_hom (1 ⊗ₜ y) x using
     2
   ·
     simp only [AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom, mul_one, one_mul,
       Algebra.TensorProduct.includeLeft_apply, Algebra.TensorProduct.tmul_mul_tmul]
-  convert(MulZeroClass.mul_zero _).symm
+  convert (MulZeroClass.mul_zero _).symm
   rw [Polynomial.eval₂_map, Algebra.TensorProduct.includeLeft_comp_algebraMap, ←
     Polynomial.eval₂_map]
   convert Polynomial.eval₂_at_apply algebra.tensor_product.include_right.to_ring_hom y
@@ -801,7 +802,7 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
     use normalizeScaleRoots_monic p this
     rw [normalizeScaleRoots_eval₂_leadingCoeff_mul p h' f x, h, MulZeroClass.mul_zero]
   · by_cases hp : p.map f = 0
-    · apply_fun fun q => coeff q p.nat_degree  at hp 
+    · apply_fun fun q => coeff q p.nat_degree at hp 
       rw [coeff_map, coeff_zero, coeff_nat_degree] at hp 
       rw [hp, MulZeroClass.zero_mul]
       exact f.is_integral_zero
@@ -828,7 +829,7 @@ end
 section IsIntegralClosure
 
 #print IsIntegralClosure /-
-/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:394:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
 /-- `is_integral_closure A R B` is the characteristic predicate stating `A` is
 the integral closure of `R` in `B`,
 i.e. that an element of `B` is integral over `R` iff it is an element of (the image of) `A`.

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

@@ -106,7 +106,7 @@ theorem isIntegral_of_noetherian (H : IsNoetherian R A) (x : A) : IsIntegral R x
   cases' HM with N HN
   have HM : ¬M < D (N + 1) :=
     WellFounded.not_lt_min (isNoetherian_iff_wellFounded.1 H) (Set.range D) _ ⟨N + 1, rfl⟩
-  rw [← HN] at HM
+  rw [← HN] at HM 
   have HN2 : D (N + 1) ≤ D N :=
     by_contradiction fun H =>
       HM (lt_of_le_not_le (map_mono (degree_le_mono (WithBot.coe_le_coe.2 (Nat.le_succ N)))) H)
@@ -156,7 +156,7 @@ theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R]
     [CommRing T] [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
     (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
     IsIntegral T (ψ a) := by
-  rw [IsIntegral, RingHom.IsIntegralElem] at ha⊢
+  rw [IsIntegral, RingHom.IsIntegralElem] at ha ⊢
   obtain ⟨p, hp⟩ := ha
   refine' ⟨p.map φ, hp.left.map _, _⟩
   rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply, eval_map, hp.right,
@@ -170,7 +170,7 @@ theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Al
   rintro ⟨p, hp, hx⟩
   use p, hp
   rwa [← f.comp_algebra_map, ← AlgHom.coe_toRingHom, ← Polynomial.hom_eval₂, AlgHom.coe_toRingHom,
-    map_eq_zero_iff f hf] at hx
+    map_eq_zero_iff f hf] at hx 
 #align is_integral_alg_hom_iff isIntegral_algHom_iff
 
 @[simp]
@@ -225,25 +225,25 @@ theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rcases hx with ⟨f, hfm, hfx⟩
   exists Finset.image ((· ^ ·) x) (Finset.range (nat_degree f + 1))
   apply le_antisymm
-  · rw [span_le]; intro s hs; rw [Finset.mem_coe] at hs
+  · rw [span_le]; intro s hs; rw [Finset.mem_coe] at hs 
     rcases Finset.mem_image.1 hs with ⟨k, hk, rfl⟩; clear hk
     exact (Algebra.adjoin R {x}).pow_mem (Algebra.subset_adjoin (Set.mem_singleton _)) k
-  intro r hr; change r ∈ Algebra.adjoin R ({x} : Set A) at hr
-  rw [Algebra.adjoin_singleton_eq_range_aeval] at hr
+  intro r hr; change r ∈ Algebra.adjoin R ({x} : Set A) at hr 
+  rw [Algebra.adjoin_singleton_eq_range_aeval] at hr 
   rcases(aeval x).mem_range.mp hr with ⟨p, rfl⟩
   rw [← mod_by_monic_add_div p hfm]
-  rw [← aeval_def] at hfx
+  rw [← aeval_def] at hfx 
   rw [AlgHom.map_add, AlgHom.map_mul, hfx, MulZeroClass.zero_mul, add_zero]
   have : degree (p %ₘ f) ≤ degree f := degree_mod_by_monic_le p hfm
-  generalize p %ₘ f = q at this⊢
+  generalize p %ₘ f = q at this ⊢
   rw [← sum_C_mul_X_pow_eq q, aeval_def, eval₂_sum, sum_def]
   refine' sum_mem fun k hkq => _
   rw [eval₂_mul, eval₂_C, eval₂_pow, eval₂_X, ← Algebra.smul_def]
   refine' smul_mem _ _ (subset_span _)
   rw [Finset.mem_coe]; refine' Finset.mem_image.2 ⟨_, _, rfl⟩
   rw [Finset.mem_range, Nat.lt_succ_iff]; refine' le_of_not_lt fun hk => _
-  rw [degree_le_iff_coeff_zero] at this
-  rw [mem_support_iff] at hkq; apply hkq; apply this
+  rw [degree_le_iff_coeff_zero] at this 
+  rw [mem_support_iff] at hkq ; apply hkq; apply this
   exact lt_of_le_of_lt degree_le_nat_degree (WithBot.coe_lt_coe.2 hk)
 #align fg_adjoin_singleton_of_integral FG_adjoin_singleton_of_integral
 
@@ -281,7 +281,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   cases' HS with y hy
   -- We can write `x` as `∑ rᵢ yᵢ` for `yᵢ ∈ Y`.
   obtain ⟨lx, hlx1, hlx2⟩ :
-    ∃ (l : A →₀ R)(H : l ∈ Finsupp.supported R R ↑y), (Finsupp.total A A R id) l = x := by
+    ∃ (l : A →₀ R) (H : l ∈ Finsupp.supported R R ↑y), (Finsupp.total A A R id) l = x := by
     rwa [← @Finsupp.mem_span_image_iff_total A A R _ _ _ id (↑y) x, Set.image_id ↑y, hy]
   -- Note that `y ⊆ S`.
   have hyS : ∀ {p}, p ∈ y → p ∈ S := fun p hp =>
@@ -289,7 +289,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   -- Now `S` is a subalgebra so the product of two elements of `y` is also in `S`.
   have : ∀ jk : (↑(y ×ˢ y) : Set (A × A)), jk.1.1 * jk.1.2 ∈ S.to_submodule := fun jk =>
     S.mul_mem (hyS (Finset.mem_product.1 jk.2).1) (hyS (Finset.mem_product.1 jk.2).2)
-  rw [← hy, ← Set.image_id ↑y] at this; simp only [Finsupp.mem_span_image_iff_total] at this
+  rw [← hy, ← Set.image_id ↑y] at this ; simp only [Finsupp.mem_span_image_iff_total] at this 
   -- Say `yᵢyⱼ = ∑rᵢⱼₖ yₖ`
   choose ly hly1 hly2
   -- Now let `S₀` be the subring of `R` generated by the `rᵢ` and the `rᵢⱼₖ`.
@@ -337,7 +337,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
     ·
       exact
         (span_le.2 (Set.insert_subset.2 ⟨(Algebra.adjoin S₀ ↑y).one_mem, Algebra.subset_adjoin⟩)) hz
-    · rw [Subalgebra.mem_toSubmodule, Algebra.mem_adjoin_iff] at hz
+    · rw [Subalgebra.mem_toSubmodule, Algebra.mem_adjoin_iff] at hz 
       suffices Subring.closure (Set.range ⇑(algebraMap (↥S₀) A) ∪ ↑y) ≤ S₁ by exact this hz
       refine' Subring.closure_le.2 (Set.union_subset _ fun t ht => subset_span <| Or.inr ht)
       rw [Set.range_subset_iff]
@@ -356,7 +356,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   have : lx r ∈ S₀ :=
     Subring.subset_closure (Finset.mem_union_left _ (Finset.mem_image_of_mem _ hr))
   change (⟨_, this⟩ : S₀) • r ∈ _
-  rw [Finsupp.mem_supported] at hlx1
+  rw [Finsupp.mem_supported] at hlx1 
   exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _
 #align is_integral_of_mem_of_fg isIntegral_of_mem_of_FG
 
@@ -385,7 +385,7 @@ theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R
         map_smul' := fun r s => LinearMap.ext fun n => Subtype.ext <| smul_assoc r s n }
       (LinearMap.ext fun n => Subtype.ext <| one_smul _ _) fun x y =>
       LinearMap.ext fun n => Subtype.ext <| mul_smul x y n
-  obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by by_contra h'; push_neg  at h'; apply hN;
+  obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by by_contra h'; push_neg  at h' ; apply hN;
     rwa [eq_bot_iff]
   have : Function.Injective f :=
     by
@@ -433,7 +433,7 @@ theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.Fin
   by
   have :=
     h.to_finite (by delta RingHom.FiniteType; convert h'; ext; exact (Algebra.smul_def _ _).symm)
-  delta RingHom.Finite at this; convert this; ext; exact Algebra.smul_def _ _
+  delta RingHom.Finite at this ; convert this; ext; exact Algebra.smul_def _ _
 #align algebra.is_integral.finite Algebra.IsIntegral.finite
 
 theorem Algebra.IsIntegral.of_finite [h : Module.Finite R A] : Algebra.IsIntegral R A :=
@@ -455,7 +455,7 @@ theorem RingHom.is_integral_of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
   by
   letI : Algebra R S := f.to_algebra
   have := (FG_adjoin_singleton_of_integral x hx).mul (FG_adjoin_singleton_of_integral y hy)
-  rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this
+  rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this 
   exact
     isIntegral_of_mem_of_FG (Algebra.adjoin R {x, y}) this z
       (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)
@@ -602,7 +602,7 @@ theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
   let ⟨p, hpm, hpx⟩ := x.2
   ⟨p, hpm,
     Subtype.eq <| by
-      rwa [← aeval_def, Subtype.val_eq_coe, ← Subalgebra.val_apply, aeval_alg_hom_apply] at hpx⟩
+      rwa [← aeval_def, Subtype.val_eq_coe, ← Subalgebra.val_apply, aeval_alg_hom_apply] at hpx ⟩
 #align integral_closure.is_integral integralClosure.isIntegral
 
 theorem RingHom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
@@ -797,16 +797,16 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
   by
   by_cases h' : 1 ≤ p.nat_degree
   · use normalizeScaleRoots p
-    have : p ≠ 0 := fun h'' => by rw [h'', nat_degree_zero] at h'; exact Nat.not_succ_le_zero 0 h'
+    have : p ≠ 0 := fun h'' => by rw [h'', nat_degree_zero] at h' ; exact Nat.not_succ_le_zero 0 h'
     use normalizeScaleRoots_monic p this
     rw [normalizeScaleRoots_eval₂_leadingCoeff_mul p h' f x, h, MulZeroClass.mul_zero]
   · by_cases hp : p.map f = 0
-    · apply_fun fun q => coeff q p.nat_degree  at hp
-      rw [coeff_map, coeff_zero, coeff_nat_degree] at hp
+    · apply_fun fun q => coeff q p.nat_degree  at hp 
+      rw [coeff_map, coeff_zero, coeff_nat_degree] at hp 
       rw [hp, MulZeroClass.zero_mul]
       exact f.is_integral_zero
-    · rw [Nat.one_le_iff_ne_zero, Classical.not_not] at h'
-      rw [eq_C_of_nat_degree_eq_zero h', eval₂_C] at h
+    · rw [Nat.one_le_iff_ne_zero, Classical.not_not] at h' 
+      rw [eq_C_of_nat_degree_eq_zero h', eval₂_C] at h 
       suffices p.map f = 0 by exact (hp this).rec _
       rw [eq_C_of_nat_degree_eq_zero h', map_C, h, C_eq_zero]
 #align ring_hom.is_integral_elem_leading_coeff_mul RingHom.isIntegralElem_leadingCoeff_mul
@@ -816,7 +816,7 @@ then `p.leading_coeff • x : S` is integral over `R`. -/
 theorem isIntegral_leadingCoeff_smul [Algebra R S] (h : aeval x p = 0) :
     IsIntegral R (p.leadingCoeff • x) :=
   by
-  rw [aeval_def] at h
+  rw [aeval_def] at h 
   rw [Algebra.smul_def]
   exact (algebraMap R S).isIntegralElem_leadingCoeff_mul p x h
 #align is_integral_leading_coeff_smul isIntegral_leadingCoeff_smul
@@ -834,7 +834,7 @@ the integral closure of `R` in `B`,
 i.e. that an element of `B` is integral over `R` iff it is an element of (the image of) `A`.
 -/
 class IsIntegralClosure (A R B : Type _) [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B]
-  [Algebra A B] : Prop where
+    [Algebra A B] : Prop where
   algebraMap_injective : Function.Injective (algebraMap A B)
   isIntegral_iff : ∀ {x : B}, IsIntegral R x ↔ ∃ y, algebraMap A B y = x
 #align is_integral_closure IsIntegralClosure
@@ -1011,7 +1011,7 @@ theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegr
   let S : Set B := ↑(p.map <| algebraMap A B).frange
   refine' isIntegral_of_mem_of_FG (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
   refine' fg_trans (FG_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
-  · rw [Finset.mem_coe, frange, Finset.mem_image] at hx
+  · rw [Finset.mem_coe, frange, Finset.mem_image] at hx 
     rcases hx with ⟨i, _, rfl⟩
     rw [coeff_map]
     exact map_isIntegral (IsScalarTower.toAlgHom R A B) (A_int _)
@@ -1050,7 +1050,7 @@ theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A
   rcases h with ⟨p, ⟨hp, hp'⟩⟩
   refine' ⟨p, ⟨hp, _⟩⟩
   rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map, eval₂_hom, ←
-    RingHom.map_zero (algebraMap A B)] at hp'
+    RingHom.map_zero (algebraMap A B)] at hp' 
   rw [eval₂_eq_eval_map]
   exact H hp'
 #align is_integral_tower_bot_of_is_integral isIntegral_tower_bot_of_isIntegral
@@ -1072,7 +1072,7 @@ theorem isIntegral_tower_bot_of_isIntegral_field {R A B : Type _} [CommRing R] [
 theorem RingHom.isIntegralElem_of_isIntegralElem_comp {x : T} (h : (g.comp f).IsIntegralElem x) :
     g.IsIntegralElem x :=
   let ⟨p, ⟨hp, hp'⟩⟩ := h
-  ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp'⟩
+  ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp' ⟩
 #align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem_of_isIntegralElem_comp
 
 theorem RingHom.isIntegral_tower_top_of_isIntegral (h : (g.comp f).IsIntegral) : g.IsIntegral :=
@@ -1085,7 +1085,7 @@ theorem isIntegral_tower_top_of_isIntegral {x : B} (h : IsIntegral R x) : IsInte
   by
   rcases h with ⟨p, ⟨hp, hp'⟩⟩
   refine' ⟨p.map (algebraMap R A), ⟨hp.map (algebraMap R A), _⟩⟩
-  rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map] at hp'
+  rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map] at hp' 
   exact hp'
 #align is_integral_tower_top_of_is_integral isIntegral_tower_top_of_isIntegral
 
@@ -1135,7 +1135,7 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
     apply (injective_iff_map_eq_zero (algebraMap R S)).mp hRS
     have a_inv_ne_zero : a_inv ≠ 0 := right_ne_zero_of_mul (mt ha_inv.symm.trans one_ne_zero)
     refine' (mul_eq_zero.mp _).resolve_right (pow_ne_zero p.nat_degree a_inv_ne_zero)
-    rw [eval₂_eq_sum_range] at hp
+    rw [eval₂_eq_sum_range] at hp 
     rw [RingHom.map_sum, Finset.sum_mul]
     refine' (Finset.sum_congr rfl fun i hi => _).trans hp
     rw [RingHom.map_mul, mul_assoc]
@@ -1146,7 +1146,7 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
   -- Since `q(a) = 0` and `q(a) = q'(a) * a + 1`, we have `a * -q'(a) = 1`.
   -- TODO: we could use a lemma for `polynomial.div_X` here.
   rw [Finset.sum_range_succ_comm, p_monic.coeff_nat_degree, one_mul, tsub_self, pow_zero,
-    add_eq_zero_iff_eq_neg, eq_comm] at hq
+    add_eq_zero_iff_eq_neg, eq_comm] at hq 
   rw [mul_comm, neg_mul, Finset.sum_mul]
   convert hq using 2
   refine' Finset.sum_congr rfl fun i hi => _

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

@@ -39,9 +39,9 @@ Let `R` be a `comm_ring` and let `A` be an R-algebra.
 -/
 
 
-open Classical
+open scoped Classical
 
-open BigOperators Polynomial
+open scoped BigOperators Polynomial
 
 open Polynomial Submodule
 
@@ -681,7 +681,7 @@ theorem IsIntegral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n)
   ⟨isIntegral_of_pow hn, fun hx => IsIntegral.pow hx n⟩
 #align is_integral.pow_iff IsIntegral.pow_iff
 
-open TensorProduct
+open scoped TensorProduct
 
 theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x ⊗ₜ[R] y) :=
   by
@@ -745,6 +745,7 @@ theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
 #align leading_coeff_smul_normalize_scale_roots leadingCoeff_smul_normalizeScaleRoots
 -/
 
+#print normalizeScaleRoots_support /-
 theorem normalizeScaleRoots_support : (normalizeScaleRoots p).support ≤ p.support :=
   by
   intro x
@@ -754,6 +755,7 @@ theorem normalizeScaleRoots_support : (normalizeScaleRoots p).support ≤ p.supp
   intro h₁ h₂
   exact (h₂ h₁).rec _
 #align normalize_scale_roots_support normalizeScaleRoots_support
+-/
 
 #print normalizeScaleRoots_degree /-
 theorem normalizeScaleRoots_degree : (normalizeScaleRoots p).degree = p.degree :=

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

@@ -89,32 +89,14 @@ protected def Algebra.IsIntegral : Prop :=
 
 variable {R A}
 
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 theorem RingHom.is_integral_map {x : R} : f.IsIntegralElem (f x) :=
   ⟨X - C x, monic_X_sub_C _, by simp⟩
 #align ring_hom.is_integral_map RingHom.is_integral_map
 
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 theorem isIntegral_algebraMap {x : R} : IsIntegral R (algebraMap R A x) :=
   (algebraMap R A).is_integral_map
 #align is_integral_algebra_map isIntegral_algebraMap
 
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-Case conversion may be inaccurate. Consider using '#align is_integral_of_noetherian isIntegral_of_noetherianₓ'. -/
 theorem isIntegral_of_noetherian (H : IsNoetherian R A) (x : A) : IsIntegral R x :=
   by
   let leval : R[X] →ₗ[R] A := (aeval x).toLinearMap
@@ -135,9 +117,6 @@ theorem isIntegral_of_noetherian (H : IsNoetherian R A) (x : A) : IsIntegral R x
   rw [LinearMap.map_sub, hpe, sub_self]
 #align is_integral_of_noetherian isIntegral_of_noetherian
 
-/- warning: is_integral_of_submodule_noetherian -> isIntegral_of_submodule_noetherian is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_of_submodule_noetherian isIntegral_of_submodule_noetherianₓ'. -/
 theorem isIntegral_of_submodule_noetherian (S : Subalgebra R A) (H : IsNoetherian R S.toSubmodule)
     (x : A) (hx : x ∈ S) : IsIntegral R x :=
   by
@@ -163,9 +142,6 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S]
 
 variable [Algebra R A] [Algebra R B] (f : R →+* S)
 
-/- warning: map_is_integral -> map_isIntegral is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align map_is_integral map_isIntegralₓ'. -/
 theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
     [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B} [AlgHomClass F A B C] (f : F)
     (hb : IsIntegral R b) : IsIntegral R (f b) :=
@@ -176,9 +152,6 @@ theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra
     aeval_alg_hom_apply, aeval_map_algebra_map, aeval_def, hP.2, _root_.map_zero]
 #align map_is_integral map_isIntegral
 
-/- warning: is_integral_map_of_comp_eq_of_is_integral -> isIntegral_map_of_comp_eq_of_isIntegral is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_map_of_comp_eq_of_is_integral isIntegral_map_of_comp_eq_of_isIntegralₓ'. -/
 theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R] [CommRing S]
     [CommRing T] [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
     (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
@@ -190,9 +163,6 @@ theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R]
     RingHom.map_zero]
 #align is_integral_map_of_comp_eq_of_is_integral isIntegral_map_of_comp_eq_of_isIntegral
 
-/- warning: is_integral_alg_hom_iff -> isIntegral_algHom_iff is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_alg_hom_iff isIntegral_algHom_iffₓ'. -/
 theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A →ₐ[R] B) (hf : Function.Injective f) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
   by
@@ -203,51 +173,27 @@ theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Al
     map_eq_zero_iff f hf] at hx
 #align is_integral_alg_hom_iff isIntegral_algHom_iff
 
-/- warning: is_integral_alg_equiv -> isIntegral_algEquiv is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_alg_equiv isIntegral_algEquivₓ'. -/
 @[simp]
 theorem isIntegral_algEquiv {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
   ⟨fun h => by simpa using map_isIntegral f.symm.to_alg_hom h, map_isIntegral f.toAlgHom⟩
 #align is_integral_alg_equiv isIntegral_algEquiv
 
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u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6)))))] {x : B}, (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_6 x) -> (IsIntegral.{u2, u3} A B _inst_2 (CommRing.toRing.{u3} B _inst_3) _inst_7 x)
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 theorem isIntegral_of_isScalarTower [Algebra A B] [IsScalarTower R A B] {x : B}
     (hx : IsIntegral R x) : IsIntegral A x :=
   let ⟨p, hp, hpx⟩ := hx
   ⟨p.map <| algebraMap R A, hp.map _, by rw [← aeval_def, aeval_map_algebra_map, aeval_def, hpx]⟩
 #align is_integral_of_is_scalar_tower isIntegral_of_isScalarTower
 
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 theorem map_isIntegral_int {B C F : Type _} [Ring B] [Ring C] {b : B} [RingHomClass F B C] (f : F)
     (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
   map_isIntegral (f : B →+* C).toIntAlgHom hb
 #align map_is_integral_int map_isIntegral_int
 
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 theorem isIntegral_ofSubring {x : A} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
   isIntegral_of_isScalarTower hx
 #align is_integral_of_subring isIntegral_ofSubring
 
-/- warning: is_integral.algebra_map -> IsIntegral.algebraMap is a dubious translation:
-<too large>
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 theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A} (h : IsIntegral R x) :
     IsIntegral R (algebraMap A B x) :=
   by
@@ -256,21 +202,12 @@ theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A} (h : I
   rw [IsScalarTower.algebraMap_eq R A B, ← hom_eval₂, hx, RingHom.map_zero]
 #align is_integral.algebra_map IsIntegral.algebraMap
 
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-<too large>
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 theorem isIntegral_algebraMap_iff [Algebra A B] [IsScalarTower R A B] {x : A}
     (hAB : Function.Injective (algebraMap A B)) :
     IsIntegral R (algebraMap A B x) ↔ IsIntegral R x :=
   isIntegral_algHom_iff (IsScalarTower.toAlgHom R A B) hAB
 #align is_integral_algebra_map_iff isIntegral_algebraMap_iff
 
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 theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
     IsIntegral R r ↔ ∃ s : Set R, s.Finite ∧ IsIntegral (Subring.closure s) r :=
   by
@@ -282,9 +219,6 @@ theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
   exact isIntegral_ofSubring _ hsr
 #align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finite
 
-/- warning: fg_adjoin_singleton_of_integral -> FG_adjoin_singleton_of_integral is a dubious translation:
-<too large>
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 theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
     (Algebra.adjoin R ({x} : Set A)).toSubmodule.FG :=
   by
@@ -313,9 +247,6 @@ theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   exact lt_of_le_of_lt degree_le_nat_degree (WithBot.coe_lt_coe.2 hk)
 #align fg_adjoin_singleton_of_integral FG_adjoin_singleton_of_integral
 
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-<too large>
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 theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
     (Algebra.adjoin R s).toSubmodule.FG :=
   Set.Finite.induction_on hfs
@@ -334,17 +265,11 @@ theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
     his
 #align fg_adjoin_of_finite FG_adjoin_of_finite
 
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 theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
     (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (↑s : Set A)) :=
   isNoetherian_of_fg_of_noetherian _ (FG_adjoin_of_finite s.finite_toSet hs)
 #align is_noetherian_adjoin_finset isNoetherian_adjoin_finset
 
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 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
   then all elements of `S` are integral over `R`. -/
@@ -435,20 +360,11 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _
 #align is_integral_of_mem_of_fg isIntegral_of_mem_of_FG
 
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 theorem Module.End.isIntegral {M : Type _} [AddCommGroup M] [Module R M] [Module.Finite R M] :
     Algebra.IsIntegral R (Module.End R M) :=
   LinearMap.exists_monic_and_aeval_eq_zero R
 #align module.End.is_integral Module.End.isIntegral
 
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-<too large>
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 /-- Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
 and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. -/
 theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R M] [Module A M]
@@ -486,32 +402,14 @@ theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R
 
 variable {f}
 
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 theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
   letI := f.to_algebra
   fun x => isIntegral_of_mem_of_FG ⊤ h.1 _ trivial
 #align ring_hom.finite.to_is_integral RingHom.Finite.to_isIntegral
 
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 alias RingHom.Finite.to_isIntegral ← RingHom.IsIntegral.of_finite
 #align ring_hom.is_integral.of_finite RingHom.IsIntegral.of_finite
 
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 theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite :=
   by
   letI := f.to_algebra
@@ -522,32 +420,14 @@ theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.
   exact FG_adjoin_of_finite (Set.toFinite _) fun x _ => h x
 #align ring_hom.is_integral.to_finite RingHom.IsIntegral.to_finite
 
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 alias RingHom.IsIntegral.to_finite ← RingHom.Finite.of_isIntegral_of_finiteType
 #align ring_hom.finite.of_is_integral_of_finite_type RingHom.Finite.of_isIntegral_of_finiteType
 
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-Case conversion may be inaccurate. Consider using '#align ring_hom.finite_iff_is_integral_and_finite_type RingHom.finite_iff_isIntegral_and_finiteTypeₓ'. -/
 /-- finite = integral + finite type -/
 theorem RingHom.finite_iff_isIntegral_and_finiteType : f.Finite ↔ f.IsIntegral ∧ f.FiniteType :=
   ⟨fun h => ⟨h.to_isIntegral, h.to_finiteType⟩, fun ⟨h, h'⟩ => h.toFinite h'⟩
 #align ring_hom.finite_iff_is_integral_and_finite_type RingHom.finite_iff_isIntegral_and_finiteType
 
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 theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.FiniteType R A] :
     Module.Finite R A :=
   by
@@ -556,24 +436,12 @@ theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.Fin
   delta RingHom.Finite at this; convert this; ext; exact Algebra.smul_def _ _
 #align algebra.is_integral.finite Algebra.IsIntegral.finite
 
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 theorem Algebra.IsIntegral.of_finite [h : Module.Finite R A] : Algebra.IsIntegral R A :=
   by
   apply RingHom.Finite.to_isIntegral
   delta RingHom.Finite; convert h; ext; exact (Algebra.smul_def _ _).symm
 #align algebra.is_integral.of_finite Algebra.IsIntegral.of_finite
 
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 /-- finite = integral + finite type -/
 theorem Algebra.finite_iff_isIntegral_and_finiteType :
     Module.Finite R A ↔ Algebra.IsIntegral R A ∧ Algebra.FiniteType R A :=
@@ -582,12 +450,6 @@ theorem Algebra.finite_iff_isIntegral_and_finiteType :
 
 variable (f)
 
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 theorem RingHom.is_integral_of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
     (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z :=
   by
@@ -599,140 +461,62 @@ theorem RingHom.is_integral_of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
       (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)
 #align ring_hom.is_integral_of_mem_closure RingHom.is_integral_of_mem_closure
 
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 theorem isIntegral_of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
     (hz : z ∈ Subring.closure ({x, y} : Set A)) : IsIntegral R z :=
   (algebraMap R A).is_integral_of_mem_closure hx hy hz
 #align is_integral_of_mem_closure isIntegral_of_mem_closure
 
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 theorem RingHom.is_integral_zero : f.IsIntegralElem 0 :=
   f.map_zero ▸ f.is_integral_map
 #align ring_hom.is_integral_zero RingHom.is_integral_zero
 
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 theorem isIntegral_zero : IsIntegral R (0 : A) :=
   (algebraMap R A).is_integral_zero
 #align is_integral_zero isIntegral_zero
 
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 theorem RingHom.is_integral_one : f.IsIntegralElem 1 :=
   f.map_one ▸ f.is_integral_map
 #align ring_hom.is_integral_one RingHom.is_integral_one
 
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 theorem isIntegral_one : IsIntegral R (1 : A) :=
   (algebraMap R A).is_integral_one
 #align is_integral_one isIntegral_one
 
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 theorem RingHom.is_integral_add {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x + y) :=
   f.is_integral_of_mem_closure hx hy <|
     Subring.add_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl))
 #align ring_hom.is_integral_add RingHom.is_integral_add
 
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 theorem isIntegral_add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x + y) :=
   (algebraMap R A).is_integral_add hx hy
 #align is_integral_add isIntegral_add
 
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 theorem RingHom.is_integral_neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) :=
   f.is_integral_of_mem_closure hx hx (Subring.neg_mem _ (Subring.subset_closure (Or.inl rfl)))
 #align ring_hom.is_integral_neg RingHom.is_integral_neg
 
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 theorem isIntegral_neg {x : A} (hx : IsIntegral R x) : IsIntegral R (-x) :=
   (algebraMap R A).is_integral_neg hx
 #align is_integral_neg isIntegral_neg
 
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 theorem RingHom.is_integral_sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x - y) := by
   simpa only [sub_eq_add_neg] using f.is_integral_add hx (f.is_integral_neg hy)
 #align ring_hom.is_integral_sub RingHom.is_integral_sub
 
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 theorem isIntegral_sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x - y) :=
   (algebraMap R A).is_integral_sub hx hy
 #align is_integral_sub isIntegral_sub
 
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 theorem RingHom.is_integral_mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x * y) :=
   f.is_integral_of_mem_closure hx hy
     (Subring.mul_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl)))
 #align ring_hom.is_integral_mul RingHom.is_integral_mul
 
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 theorem isIntegral_mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x * y) :=
   (algebraMap R A).is_integral_mul hx hy
@@ -747,12 +531,6 @@ theorem isIntegral_smul [Algebra S A] [Algebra R S] [IsScalarTower R S A] {x : A
 #align is_integral_smul isIntegral_smul
 -/
 
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 theorem isIntegral_of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
     IsIntegral R x := by
   rcases hx with ⟨p, ⟨hmonic, heval⟩⟩
@@ -776,9 +554,6 @@ def integralClosure : Subalgebra R A
 #align integral_closure integralClosure
 -/
 
-/- warning: mem_integral_closure_iff_mem_fg -> mem_integralClosure_iff_mem_FG is a dubious translation:
-<too large>
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 theorem mem_integralClosure_iff_mem_FG {r : A} :
     r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M :=
   ⟨fun hr =>
@@ -788,12 +563,6 @@ theorem mem_integralClosure_iff_mem_FG {r : A} :
 
 variable {R} {A}
 
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 theorem adjoin_le_integralClosure {x : A} (hx : IsIntegral R x) :
     Algebra.adjoin R {x} ≤ integralClosure R A :=
   by
@@ -802,12 +571,6 @@ theorem adjoin_le_integralClosure {x : A} (hx : IsIntegral R x) :
   exact hx
 #align adjoin_le_integral_closure adjoin_le_integralClosure
 
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 theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
     S ≤ integralClosure R A ↔ Algebra.IsIntegral R S :=
   SetLike.forall.symm.trans
@@ -816,17 +579,11 @@ theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
         isIntegral_algebraMap_iff Subtype.coe_injective)
 #align le_integral_closure_iff_is_integral le_integralClosure_iff_isIntegral
 
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 theorem isIntegral_sup {S T : Subalgebra R A} :
     Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R S ∧ Algebra.IsIntegral R T := by
   simp only [← le_integralClosure_iff_isIntegral, sup_le_iff]
 #align is_integral_sup isIntegral_sup
 
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 /-- Mapping an integral closure along an `alg_equiv` gives the integral closure. -/
 theorem integralClosure_map_algEquiv (f : A ≃ₐ[R] B) :
     (integralClosure R A).map (f : A →ₐ[R] B) = integralClosure R B :=
@@ -841,12 +598,6 @@ theorem integralClosure_map_algEquiv (f : A ≃ₐ[R] B) :
     simp
 #align integral_closure_map_alg_equiv integralClosure_map_algEquiv
 
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 theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
   let ⟨p, hpm, hpx⟩ := x.2
   ⟨p, hpm,
@@ -854,12 +605,6 @@ theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
       rwa [← aeval_def, Subtype.val_eq_coe, ← Subalgebra.val_apply, aeval_alg_hom_apply] at hpx⟩
 #align integral_closure.is_integral integralClosure.isIntegral
 
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 theorem RingHom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
     (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x :=
   by
@@ -869,23 +614,11 @@ theorem RingHom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r
   rw [mul_comm x y, ← mul_assoc, hr, one_mul]
 #align ring_hom.is_integral_of_is_integral_mul_unit RingHom.is_integral_of_is_integral_mul_unit
 
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 theorem isIntegral_of_isIntegral_mul_unit {x y : A} {r : R} (hr : algebraMap R A r * y = 1)
     (hx : IsIntegral R (x * y)) : IsIntegral R x :=
   (algebraMap R A).is_integral_of_is_integral_mul_unit x y r hr hx
 #align is_integral_of_is_integral_mul_unit isIntegral_of_isIntegral_mul_unit
 
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 /-- Generalization of `is_integral_of_mem_closure` bootstrapped up from that lemma -/
 theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) :
     ∀ x ∈ Subring.closure G, IsIntegral R x := fun x hx =>
@@ -893,43 +626,19 @@ theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x
     (fun _ => isIntegral_neg) fun _ _ => isIntegral_mul
 #align is_integral_of_mem_closure' isIntegral_of_mem_closure'
 
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 theorem is_integral_of_mem_closure'' {S : Type _} [CommRing S] {f : R →+* S} (G : Set S)
     (hG : ∀ x ∈ G, f.IsIntegralElem x) : ∀ x ∈ Subring.closure G, f.IsIntegralElem x := fun x hx =>
   @isIntegral_of_mem_closure' R S _ _ f.toAlgebra G hG x hx
 #align is_integral_of_mem_closure'' is_integral_of_mem_closure''
 
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 theorem IsIntegral.pow {x : A} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (x ^ n) :=
   (integralClosure R A).pow_mem h n
 #align is_integral.pow IsIntegral.pow
 
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 theorem IsIntegral.nsmul {x : A} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (n • x) :=
   (integralClosure R A).nsmul_mem h n
 #align is_integral.nsmul IsIntegral.nsmul
 
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 theorem IsIntegral.zsmul {x : A} (h : IsIntegral R x) (n : ℤ) : IsIntegral R (n • x) :=
   (integralClosure R A).zsmul_mem h n
 #align is_integral.zsmul IsIntegral.zsmul
@@ -948,23 +657,11 @@ theorem IsIntegral.multiset_sum {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R
 #align is_integral.multiset_sum IsIntegral.multiset_sum
 -/
 
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 theorem IsIntegral.prod {α : Type _} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
     IsIntegral R (∏ x in s, f x) :=
   (integralClosure R A).prod_mem h
 #align is_integral.prod IsIntegral.prod
 
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 theorem IsIntegral.sum {α : Type _} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
     IsIntegral R (∑ x in s, f x) :=
   (integralClosure R A).sum_mem h
@@ -979,12 +676,6 @@ theorem IsIntegral.det {n : Type _} [Fintype n] [DecidableEq n] {M : Matrix n n
 #align is_integral.det IsIntegral.det
 -/
 
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 @[simp]
 theorem IsIntegral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n) ↔ IsIntegral R x :=
   ⟨isIntegral_of_pow hn, fun hx => IsIntegral.pow hx n⟩
@@ -992,12 +683,6 @@ theorem IsIntegral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n)
 
 open TensorProduct
 
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-Case conversion may be inaccurate. Consider using '#align is_integral.tmul IsIntegral.tmulₓ'. -/
 theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x ⊗ₜ[R] y) :=
   by
   obtain ⟨p, hp, hp'⟩ := h
@@ -1028,12 +713,6 @@ noncomputable def normalizeScaleRoots (p : R[X]) : R[X] :=
 #align normalize_scale_roots normalizeScaleRoots
 -/
 
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-Case conversion may be inaccurate. Consider using '#align normalize_scale_roots_coeff_mul_leading_coeff_pow normalizeScaleRoots_coeff_mul_leadingCoeff_powₓ'. -/
 theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ natDegree p) :
     (normalizeScaleRoots p).coeff i * p.leadingCoeff ^ i =
       p.coeff i * p.leadingCoeff ^ (p.natDegree - 1) :=
@@ -1066,12 +745,6 @@ theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
 #align leading_coeff_smul_normalize_scale_roots leadingCoeff_smul_normalizeScaleRoots
 -/
 
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 theorem normalizeScaleRoots_support : (normalizeScaleRoots p).support ≤ p.support :=
   by
   intro x
@@ -1092,9 +765,6 @@ theorem normalizeScaleRoots_degree : (normalizeScaleRoots p).degree = p.degree :
 #align normalize_scale_roots_degree normalizeScaleRoots_degree
 -/
 
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-<too large>
-Case conversion may be inaccurate. Consider using '#align normalize_scale_roots_eval₂_leading_coeff_mul normalizeScaleRoots_eval₂_leadingCoeff_mulₓ'. -/
 theorem normalizeScaleRoots_eval₂_leadingCoeff_mul (h : 1 ≤ p.natDegree) (f : R →+* S) (x : S) :
     (normalizeScaleRoots p).eval₂ f (f p.leadingCoeff * x) =
       f p.leadingCoeff ^ (p.natDegree - 1) * p.eval₂ f x :=
@@ -1118,12 +788,6 @@ theorem normalizeScaleRoots_monic (h : p ≠ 0) : (normalizeScaleRoots p).Monic
 #align normalize_scale_roots_monic normalizeScaleRoots_monic
 -/
 
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-Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_elem_leading_coeff_mul RingHom.isIntegralElem_leadingCoeff_mulₓ'. -/
 /-- Given a `p : R[X]` and a `x : S` such that `p.eval₂ f x = 0`,
 `f p.leading_coeff * x` is integral. -/
 theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
@@ -1145,9 +809,6 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
       rw [eq_C_of_nat_degree_eq_zero h', map_C, h, C_eq_zero]
 #align ring_hom.is_integral_elem_leading_coeff_mul RingHom.isIntegralElem_leadingCoeff_mul
 
-/- warning: is_integral_leading_coeff_smul -> isIntegral_leadingCoeff_smul is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_leading_coeff_smul isIntegral_leadingCoeff_smulₓ'. -/
 /-- Given a `p : R[X]` and a root `x : S`,
 then `p.leading_coeff • x : S` is integral over `R`. -/
 theorem isIntegral_leadingCoeff_smul [Algebra R S] (h : aeval x p = 0) :
@@ -1177,12 +838,6 @@ class IsIntegralClosure (A R B : Type _) [CommRing R] [CommSemiring A] [CommRing
 #align is_integral_closure IsIntegralClosure
 -/
 
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-Case conversion may be inaccurate. Consider using '#align integral_closure.is_integral_closure integralClosure.isIntegralClosureₓ'. -/
 instance integralClosure.isIntegralClosure (R A : Type _) [CommRing R] [CommRing A] [Algebra R A] :
     IsIntegralClosure (integralClosure R A) R A :=
   ⟨Subtype.coe_injective, fun x => ⟨fun h => ⟨⟨x, h⟩, rfl⟩, by rintro ⟨⟨_, h⟩, rfl⟩; exact h⟩⟩
@@ -1196,30 +851,15 @@ variable [Algebra R B] [Algebra A B] [IsIntegralClosure A R B]
 
 variable (R) {A} (B)
 
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(Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] (x : A), IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_7 x
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-Case conversion may be inaccurate. Consider using '#align is_integral_closure.is_integral IsIntegralClosure.isIntegralₓ'. -/
 protected theorem isIntegral [Algebra R A] [IsScalarTower R A B] (x : A) : IsIntegral R x :=
   (isIntegral_algebraMap_iff (algebraMap_injective A R B)).mp <|
     show IsIntegral R (algebraMap A B x) from isIntegral_iff.mpr ⟨x, rfl⟩
 #align is_integral_closure.is_integral IsIntegralClosure.isIntegral
 
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A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_7))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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)))))) 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(Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))], Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_7
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-Case conversion may be inaccurate. Consider using '#align is_integral_closure.is_integral_algebra IsIntegralClosure.isIntegral_algebraₓ'. -/
 theorem isIntegral_algebra [Algebra R A] [IsScalarTower R A B] : Algebra.IsIntegral R A := fun x =>
   IsIntegralClosure.isIntegral R B x
 #align is_integral_closure.is_integral_algebra IsIntegralClosure.isIntegral_algebra
 
-/- warning: is_integral_closure.no_zero_smul_divisors -> IsIntegralClosure.noZeroSMulDivisors is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_closure.no_zero_smul_divisors IsIntegralClosure.noZeroSMulDivisorsₓ'. -/
 theorem noZeroSMulDivisors [Algebra R A] [IsScalarTower R A B] [NoZeroSMulDivisors R B] :
     NoZeroSMulDivisors R A :=
   by
@@ -1238,66 +878,33 @@ noncomputable def mk' (x : B) (hx : IsIntegral R x) : A :=
 #align is_integral_closure.mk' IsIntegralClosure.mk'
 -/
 
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 @[simp]
 theorem algebraMap_mk' (x : B) (hx : IsIntegral R x) : algebraMap A B (mk' A x hx) = x :=
   Classical.choose_spec (isIntegral_iff.mp hx)
 #align is_integral_closure.algebra_map_mk' IsIntegralClosure.algebraMap_mk'
 
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 @[simp]
 theorem mk'_one (h : IsIntegral R (1 : B) := isIntegral_one) : mk' A 1 h = 1 :=
   algebraMap_injective A R B <| by rw [algebra_map_mk', RingHom.map_one]
 #align is_integral_closure.mk'_one IsIntegralClosure.mk'_one
 
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 @[simp]
 theorem mk'_zero (h : IsIntegral R (0 : B) := isIntegral_zero) : mk' A 0 h = 0 :=
   algebraMap_injective A R B <| by rw [algebra_map_mk', RingHom.map_zero]
 #align is_integral_closure.mk'_zero IsIntegralClosure.mk'_zero
 
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 @[simp]
 theorem mk'_add (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
     mk' A (x + y) (isIntegral_add hx hy) = mk' A x hx + mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebra_map_mk', RingHom.map_add]
 #align is_integral_closure.mk'_add IsIntegralClosure.mk'_add
 
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-  forall {R : Type.{u1}} (A : Type.{u2}) {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] (x : B) (y : B) (hx : IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_4 x) (hy : IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_4 y), Eq.{succ u2} A (IsIntegralClosure.mk'.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 (HMul.hMul.{u3, u3, u3} B B B (instHMul.{u3} B (Distrib.toHasMul.{u3} B (Ring.toDistrib.{u3} B (CommRing.toRing.{u3} B _inst_3)))) x y) (isIntegral_mul.{u1, u3} R B _inst_1 _inst_3 _inst_4 x y hx hy)) (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (Distrib.toHasMul.{u2} A (Ring.toDistrib.{u2} A (CommRing.toRing.{u2} A _inst_2)))) (IsIntegralClosure.mk'.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 x hx) (IsIntegralClosure.mk'.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 y hy))
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-  forall {R : Type.{u3}} (A : Type.{u1}) {B : Type.{u2}} [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : CommRing.{u2} B] [_inst_4 : Algebra.{u3, u2} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_5 : Algebra.{u1, u2} A B (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_6 : IsIntegralClosure.{u1, u3, u2} A R B _inst_1 (CommRing.toCommSemiring.{u1} A _inst_2) _inst_3 _inst_4 _inst_5] (x : B) (y : B) (hx : IsIntegral.{u3, u2} R B _inst_1 (CommRing.toRing.{u2} B _inst_3) _inst_4 x) (hy : IsIntegral.{u3, u2} R B _inst_1 (CommRing.toRing.{u2} B _inst_3) _inst_4 y), Eq.{succ u1} A (IsIntegralClosure.mk'.{u3, u1, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 (HMul.hMul.{u2, u2, u2} B B B (instHMul.{u2} B (NonUnitalNonAssocRing.toMul.{u2} B (NonAssocRing.toNonUnitalNonAssocRing.{u2} B (Ring.toNonAssocRing.{u2} B (CommRing.toRing.{u2} B _inst_3))))) x y) (isIntegral_mul.{u2, u3} R B _inst_1 _inst_3 _inst_4 x y hx hy)) (HMul.hMul.{u1, u1, u1} A A A (instHMul.{u1} A (NonUnitalNonAssocRing.toMul.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (IsIntegralClosure.mk'.{u3, u1, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 x hx) (IsIntegralClosure.mk'.{u3, u1, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 y hy))
-Case conversion may be inaccurate. Consider using '#align is_integral_closure.mk'_mul IsIntegralClosure.mk'_mulₓ'. -/
 @[simp]
 theorem mk'_mul (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
     mk' A (x * y) (isIntegral_mul hx hy) = mk' A x hx * mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebra_map_mk', RingHom.map_mul]
 #align is_integral_closure.mk'_mul IsIntegralClosure.mk'_mul
 
-/- warning: is_integral_closure.mk'_algebra_map -> IsIntegralClosure.mk'_algebraMap is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_closure.mk'_algebra_map IsIntegralClosure.mk'_algebraMapₓ'. -/
 @[simp]
 theorem mk'_algebraMap [Algebra R A] [IsScalarTower R A B] (x : R)
     (h : IsIntegral R (algebraMap R B x) := isIntegral_algebraMap) :
@@ -1325,9 +932,6 @@ noncomputable def lift : S →ₐ[R] A
 #align is_integral_closure.lift IsIntegralClosure.lift
 -/
 
-/- warning: is_integral_closure.algebra_map_lift -> IsIntegralClosure.algebraMap_lift is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_closure.algebra_map_lift IsIntegralClosure.algebraMap_liftₓ'. -/
 @[simp]
 theorem algebraMap_lift (x : S) : algebraMap A B (lift A B h x) = algebraMap S B x :=
   algebraMap_mk' _ _ _
@@ -1350,9 +954,6 @@ noncomputable def equiv : A ≃ₐ[R] A' :=
 #align is_integral_closure.equiv IsIntegralClosure.equiv
 -/
 
-/- warning: is_integral_closure.algebra_map_equiv -> IsIntegralClosure.algebraMap_equiv is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_closure.algebra_map_equiv IsIntegralClosure.algebraMap_equivₓ'. -/
 @[simp]
 theorem algebraMap_equiv (x : A) : algebraMap A' B (equiv R A B A' x) = algebraMap A B x :=
   algebraMap_lift _ _ _ _
@@ -1374,9 +975,6 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S] [CommRing T]
 
 variable [Algebra A B] [Algebra R B] (f : R →+* S) (g : S →+* T)
 
-/- warning: is_integral_trans_aux -> isIntegral_trans_aux is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_trans_aux isIntegral_trans_auxₓ'. -/
 theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x p = 0) :
     IsIntegral (adjoin R (↑(p.map <| algebraMap A B).frange : Set B)) x :=
   by
@@ -1403,12 +1001,6 @@ theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x
 
 variable [Algebra R A] [IsScalarTower R A B]
 
-/- warning: is_integral_trans -> isIntegral_trans is a dubious translation:
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(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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A 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u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7)))))], (Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8) -> (forall (x : B), (IsIntegral.{u2, u3} A B _inst_2 (CommRing.toRing.{u3} B _inst_3) _inst_6 x) -> (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_7 x))
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-Case conversion may be inaccurate. Consider using '#align is_integral_trans isIntegral_transₓ'. -/
 /-- If A is an R-algebra all of whose elements are integral over R,
 and x is an element of an A-algebra that is integral over A, then x is integral over R.-/
 theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegral A x) :
@@ -1425,12 +1017,6 @@ theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegr
     exact isIntegral_trans_aux _ pmonic hp
 #align is_integral_trans isIntegral_trans
 
-/- warning: algebra.is_integral_trans -> Algebra.isIntegral_trans is a dubious translation:
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u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7)))))], (Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8) -> (Algebra.IsIntegral.{u2, u3} A B _inst_2 (CommRing.toRing.{u3} B _inst_3) _inst_6) -> (Algebra.IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_7)
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 /-- If A is an R-algebra all of whose elements are integral over R,
 and B is an A-algebra all of whose elements are integral over A,
 then all elements of B are integral over R.-/
@@ -1438,12 +1024,6 @@ theorem Algebra.isIntegral_trans (hA : Algebra.IsIntegral R A) (hB : Algebra.IsI
     Algebra.IsIntegral R B := fun x => isIntegral_trans hA x (hB x)
 #align algebra.is_integral_trans Algebra.isIntegral_trans
 
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 theorem RingHom.isIntegral_trans (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.comp f).IsIntegral :=
   @Algebra.isIntegral_trans R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
     (@IsScalarTower.of_algebraMap_eq R S T _ _ _ f.toAlgebra g.toAlgebra (g.comp f).toAlgebra
@@ -1451,30 +1031,15 @@ theorem RingHom.isIntegral_trans (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.co
     hf hg
 #align ring_hom.is_integral_trans RingHom.isIntegral_trans
 
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 theorem RingHom.isIntegral_of_surjective (hf : Function.Surjective f) : f.IsIntegral := fun x =>
   (hf x).recOn fun y hy => (hy ▸ f.is_integral_map : f.IsIntegralElem x)
 #align ring_hom.is_integral_of_surjective RingHom.isIntegral_of_surjective
 
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 theorem isIntegral_of_surjective (h : Function.Surjective (algebraMap R A)) :
     Algebra.IsIntegral R A :=
   (algebraMap R A).isIntegral_of_surjective h
 #align is_integral_of_surjective isIntegral_of_surjective
 
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-<too large>
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 /-- If `R → A → B` is an algebra tower with `A → B` injective,
 then if the entire tower is an integral extension so is `R → A` -/
 theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A B)) {x : A}
@@ -1488,12 +1053,6 @@ theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A
   exact H hp'
 #align is_integral_tower_bot_of_is_integral isIntegral_tower_bot_of_isIntegral
 
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 theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injective g)
     (hfg : (g.comp f).IsIntegral) : f.IsIntegral := fun x =>
   @isIntegral_tower_bot_of_isIntegral R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
@@ -1502,43 +1061,22 @@ theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injective g)
     hg x (hfg (g x))
 #align ring_hom.is_integral_tower_bot_of_is_integral RingHom.isIntegral_tower_bot_of_isIntegral
 
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 theorem isIntegral_tower_bot_of_isIntegral_field {R A B : Type _} [CommRing R] [Field A]
     [CommRing B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]
     {x : A} (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
   isIntegral_tower_bot_of_isIntegral (algebraMap A B).Injective h
 #align is_integral_tower_bot_of_is_integral_field isIntegral_tower_bot_of_isIntegral_field
 
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 theorem RingHom.isIntegralElem_of_isIntegralElem_comp {x : T} (h : (g.comp f).IsIntegralElem x) :
     g.IsIntegralElem x :=
   let ⟨p, ⟨hp, hp'⟩⟩ := h
   ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp'⟩
 #align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem_of_isIntegralElem_comp
 
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 theorem RingHom.isIntegral_tower_top_of_isIntegral (h : (g.comp f).IsIntegral) : g.IsIntegral :=
   fun x => RingHom.isIntegralElem_of_isIntegralElem_comp f g (h x)
 #align ring_hom.is_integral_tower_top_of_is_integral RingHom.isIntegral_tower_top_of_isIntegral
 
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u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7)))))] {x : B}, (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_7 x) -> (IsIntegral.{u2, u3} A B _inst_2 (CommRing.toRing.{u3} B _inst_3) _inst_6 x)
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-Case conversion may be inaccurate. Consider using '#align is_integral_tower_top_of_is_integral isIntegral_tower_top_of_isIntegralₓ'. -/
 /-- If `R → A → B` is an algebra tower,
 then if the entire tower is an integral extension so is `A → B`. -/
 theorem isIntegral_tower_top_of_isIntegral {x : B} (h : IsIntegral R x) : IsIntegral A x :=
@@ -1549,12 +1087,6 @@ theorem isIntegral_tower_top_of_isIntegral {x : B} (h : IsIntegral R x) : IsInte
   exact hp'
 #align is_integral_tower_top_of_is_integral isIntegral_tower_top_of_isIntegral
 
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-Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_quotient_of_is_integral RingHom.isIntegral_quotient_of_isIntegralₓ'. -/
 theorem RingHom.isIntegral_quotient_of_isIntegral {I : Ideal S} (hf : f.IsIntegral) :
     (Ideal.quotientMap I f le_rfl).IsIntegral :=
   by
@@ -1564,20 +1096,11 @@ theorem RingHom.isIntegral_quotient_of_isIntegral {I : Ideal S} (hf : f.IsIntegr
   simpa only [hom_eval₂, eval₂_map] using congr_arg (Ideal.Quotient.mk I) hpx
 #align ring_hom.is_integral_quotient_of_is_integral RingHom.isIntegral_quotient_of_isIntegral
 
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 theorem isIntegral_quotient_of_isIntegral {I : Ideal A} (hRA : Algebra.IsIntegral R A) :
     Algebra.IsIntegral (R ⧸ I.comap (algebraMap R A)) (A ⧸ I) :=
   (algebraMap R A).isIntegral_quotient_of_isIntegral hRA
 #align is_integral_quotient_of_is_integral isIntegral_quotient_of_isIntegral
 
-/- warning: is_integral_quotient_map_iff -> isIntegral_quotientMap_iff is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align is_integral_quotient_map_iff isIntegral_quotientMap_iffₓ'. -/
 theorem isIntegral_quotientMap_iff {I : Ideal S} :
     (Ideal.quotientMap I f le_rfl).IsIntegral ↔
       ((Ideal.Quotient.mk I).comp f : R →+* S ⧸ I).IsIntegral :=
@@ -1589,12 +1112,6 @@ theorem isIntegral_quotientMap_iff {I : Ideal S} :
   exact RingHom.isIntegral_of_surjective g Ideal.Quotient.mk_surjective
 #align is_integral_quotient_map_iff isIntegral_quotientMap_iff
 
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 /-- If the integral extension `R → S` is injective, and `S` is a field, then `R` is also a field. -/
 theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial R] [CommRing S]
     [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S)
@@ -1635,12 +1152,6 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
   rw [mul_assoc, ← pow_succ', tsub_add_cancel_of_le this]
 #align is_field_of_is_integral_of_is_field isField_of_isIntegral_of_isField
 
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-Case conversion may be inaccurate. Consider using '#align is_field_of_is_integral_of_is_field' isField_of_isIntegral_of_isField'ₓ'. -/
 theorem isField_of_isIntegral_of_isField' {R S : Type _} [CommRing R] [CommRing S] [IsDomain S]
     [Algebra R S] (H : Algebra.IsIntegral R S) (hR : IsField R) : IsField S :=
   by
@@ -1658,12 +1169,6 @@ theorem isField_of_isIntegral_of_isField' {R S : Type _} [CommRing R] [CommRing
   exact ⟨y, subtype.ext_iff.mp hy⟩
 #align is_field_of_is_integral_of_is_field' isField_of_isIntegral_of_isField'
 
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-Case conversion may be inaccurate. Consider using '#align algebra.is_integral.is_field_iff_is_field Algebra.IsIntegral.isField_iff_isFieldₓ'. -/
 theorem Algebra.IsIntegral.isField_iff_isField {R S : Type _} [CommRing R] [Nontrivial R]
     [CommRing S] [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S)
     (hRS : Function.Injective (algebraMap R S)) : IsField R ↔ IsField S :=
@@ -1672,9 +1177,6 @@ theorem Algebra.IsIntegral.isField_iff_isField {R S : Type _} [CommRing R] [Nont
 
 end Algebra
 
-/- warning: integral_closure_idem -> integralClosure_idem is a dubious translation:
-<too large>
-Case conversion may be inaccurate. Consider using '#align integral_closure_idem integralClosure_idemₓ'. -/
 theorem integralClosure_idem {R : Type _} {A : Type _} [CommRing R] [CommRing A] [Algebra R A] :
     integralClosure (integralClosure R A : Set A) A = ⊥ :=
   eq_bot_iff.2 fun x hx =>
@@ -1692,12 +1194,6 @@ variable {R S : Type _} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S]
 instance : IsDomain (integralClosure R S) :=
   inferInstance
 
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-Case conversion may be inaccurate. Consider using '#align roots_mem_integral_closure roots_mem_integralClosureₓ'. -/
 theorem roots_mem_integralClosure {f : R[X]} (hf : f.Monic) {a : S}
     (ha : a ∈ (f.map <| algebraMap R S).roots) : a ∈ integralClosure R S :=
   ⟨f, hf, (eval₂_eq_eval_map _).trans <| (mem_roots <| (hf.map _).NeZero).1 ha⟩

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

@@ -291,11 +291,8 @@ theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rcases hx with ⟨f, hfm, hfx⟩
   exists Finset.image ((· ^ ·) x) (Finset.range (nat_degree f + 1))
   apply le_antisymm
-  · rw [span_le]
-    intro s hs
-    rw [Finset.mem_coe] at hs
-    rcases Finset.mem_image.1 hs with ⟨k, hk, rfl⟩
-    clear hk
+  · rw [span_le]; intro s hs; rw [Finset.mem_coe] at hs
+    rcases Finset.mem_image.1 hs with ⟨k, hk, rfl⟩; clear hk
     exact (Algebra.adjoin R {x}).pow_mem (Algebra.subset_adjoin (Set.mem_singleton _)) k
   intro r hr; change r ∈ Algebra.adjoin R ({x} : Set A) at hr
   rw [Algebra.adjoin_singleton_eq_range_aeval] at hr
@@ -363,14 +360,11 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
     rwa [← @Finsupp.mem_span_image_iff_total A A R _ _ _ id (↑y) x, Set.image_id ↑y, hy]
   -- Note that `y ⊆ S`.
   have hyS : ∀ {p}, p ∈ y → p ∈ S := fun p hp =>
-    show p ∈ S.to_submodule by
-      rw [← hy]
-      exact subset_span hp
+    show p ∈ S.to_submodule by rw [← hy]; exact subset_span hp
   -- Now `S` is a subalgebra so the product of two elements of `y` is also in `S`.
   have : ∀ jk : (↑(y ×ˢ y) : Set (A × A)), jk.1.1 * jk.1.2 ∈ S.to_submodule := fun jk =>
     S.mul_mem (hyS (Finset.mem_product.1 jk.2).1) (hyS (Finset.mem_product.1 jk.2).2)
-  rw [← hy, ← Set.image_id ↑y] at this
-  simp only [Finsupp.mem_span_image_iff_total] at this
+  rw [← hy, ← Set.image_id ↑y] at this; simp only [Finsupp.mem_span_image_iff_total] at this
   -- Say `yᵢyⱼ = ∑rᵢⱼₖ yₖ`
   choose ly hly1 hly2
   -- Now let `S₀` be the subring of `R` generated by the `rᵢ` and the `rᵢⱼₖ`.
@@ -385,14 +379,11 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   have :
     span S₀ (insert 1 ↑y : Set A) * span S₀ (insert 1 ↑y : Set A) ≤ span S₀ (insert 1 ↑y : Set A) :=
     by
-    rw [span_mul_span]
-    refine' span_le.2 fun z hz => _
+    rw [span_mul_span]; refine' span_le.2 fun z hz => _
     rcases Set.mem_mul.1 hz with ⟨p, q, rfl | hp, hq, rfl⟩
-    · rw [one_mul]
-      exact subset_span hq
+    · rw [one_mul]; exact subset_span hq
     rcases hq with (rfl | hq)
-    · rw [mul_one]
-      exact subset_span (Or.inr hp)
+    · rw [mul_one]; exact subset_span (Or.inr hp)
     erw [← hly2 ⟨(p, q), Finset.mem_product.2 ⟨hp, hq⟩⟩]
     rw [Finsupp.total_apply, Finsupp.sum]
     refine' (span S₀ (insert 1 ↑y : Set A)).sum_mem fun t ht => _
@@ -402,8 +393,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
           Finset.mem_biUnion.2
             ⟨⟨(p, q), Finset.mem_product.2 ⟨hp, hq⟩⟩, Finset.mem_univ _,
               Finsupp.mem_frange.2 ⟨Finsupp.mem_support_iff.1 ht, _, rfl⟩⟩)
-    change (⟨_, this⟩ : S₀) • t ∈ _
-    exact smul_mem _ _ (subset_span <| Or.inr <| hly1 _ ht)
+    change (⟨_, this⟩ : S₀) • t ∈ _; exact smul_mem _ _ (subset_span <| Or.inr <| hly1 _ ht)
   -- Hence this span is a subring. Call this subring `S₁`.
   let S₁ : Subring A :=
     { carrier := span S₀ (insert 1 ↑y : Set A)
@@ -435,14 +425,9 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   refine'
     isIntegral_of_submodule_noetherian (Algebra.adjoin S₀ ↑y)
       (isNoetherian_of_fg_of_noetherian _
-        ⟨insert 1 y, by
-          rw [Finset.coe_insert]
-          ext z
-          simp [S₁]
-          convert foo z⟩)
+        ⟨insert 1 y, by rw [Finset.coe_insert]; ext z; simp [S₁]; convert foo z⟩)
       _ _
-  rw [← hlx2, Finsupp.total_apply, Finsupp.sum]
-  refine' Subalgebra.sum_mem _ fun r hr => _
+  rw [← hlx2, Finsupp.total_apply, Finsupp.sum]; refine' Subalgebra.sum_mem _ fun r hr => _
   have : lx r ∈ S₀ :=
     Subring.subset_closure (Finset.mem_union_left _ (Finset.mem_image_of_mem _ hr))
   change (⟨_, this⟩ : S₀) • r ∈ _
@@ -484,11 +469,7 @@ theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R
         map_smul' := fun r s => LinearMap.ext fun n => Subtype.ext <| smul_assoc r s n }
       (LinearMap.ext fun n => Subtype.ext <| one_smul _ _) fun x y =>
       LinearMap.ext fun n => Subtype.ext <| mul_smul x y n
-  obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) :=
-    by
-    by_contra h'
-    push_neg  at h'
-    apply hN
+  obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by by_contra h'; push_neg  at h'; apply hN;
     rwa [eq_bot_iff]
   have : Function.Injective f :=
     by
@@ -571,12 +552,7 @@ theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.Fin
     Module.Finite R A :=
   by
   have :=
-    h.to_finite
-      (by
-        delta RingHom.FiniteType
-        convert h'
-        ext
-        exact (Algebra.smul_def _ _).symm)
+    h.to_finite (by delta RingHom.FiniteType; convert h'; ext; exact (Algebra.smul_def _ _).symm)
   delta RingHom.Finite at this; convert this; ext; exact Algebra.smul_def _ _
 #align algebra.is_integral.finite Algebra.IsIntegral.finite
 
@@ -1026,8 +1002,7 @@ theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x 
   by
   obtain ⟨p, hp, hp'⟩ := h
   refine' ⟨(p.map (algebraMap R A)).scaleRoots x, _, _⟩
-  · rw [Polynomial.monic_scaleRoots_iff]
-    exact hp.map _
+  · rw [Polynomial.monic_scaleRoots_iff]; exact hp.map _
   convert@Polynomial.scaleRoots_eval₂_mul (A ⊗[R] B) A _ _ _
       algebra.tensor_product.include_left.to_ring_hom (1 ⊗ₜ y) x using
     2
@@ -1156,9 +1131,7 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
   by
   by_cases h' : 1 ≤ p.nat_degree
   · use normalizeScaleRoots p
-    have : p ≠ 0 := fun h'' => by
-      rw [h'', nat_degree_zero] at h'
-      exact Nat.not_succ_le_zero 0 h'
+    have : p ≠ 0 := fun h'' => by rw [h'', nat_degree_zero] at h'; exact Nat.not_succ_le_zero 0 h'
     use normalizeScaleRoots_monic p this
     rw [normalizeScaleRoots_eval₂_leadingCoeff_mul p h' f x, h, MulZeroClass.mul_zero]
   · by_cases hp : p.map f = 0
@@ -1212,10 +1185,7 @@ but is expected to have type
 Case conversion may be inaccurate. Consider using '#align integral_closure.is_integral_closure integralClosure.isIntegralClosureₓ'. -/
 instance integralClosure.isIntegralClosure (R A : Type _) [CommRing R] [CommRing A] [Algebra R A] :
     IsIntegralClosure (integralClosure R A) R A :=
-  ⟨Subtype.coe_injective, fun x =>
-    ⟨fun h => ⟨⟨x, h⟩, rfl⟩, by
-      rintro ⟨⟨_, h⟩, rfl⟩
-      exact h⟩⟩
+  ⟨Subtype.coe_injective, fun x => ⟨fun h => ⟨⟨x, h⟩, rfl⟩, by rintro ⟨⟨_, h⟩, rfl⟩; exact h⟩⟩
 #align integral_closure.is_integral_closure integralClosure.isIntegralClosure
 
 namespace IsIntegralClosure
@@ -1375,14 +1345,8 @@ variable [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A'
 /-- Integral closures are all isomorphic to each other. -/
 noncomputable def equiv : A ≃ₐ[R] A' :=
   AlgEquiv.ofAlgHom (lift _ B (isIntegral_algebra R B)) (lift _ B (isIntegral_algebra R B))
-    (by
-      ext x
-      apply algebra_map_injective A' R B
-      simp)
-    (by
-      ext x
-      apply algebra_map_injective A R B
-      simp)
+    (by ext x; apply algebra_map_injective A' R B; simp)
+    (by ext x; apply algebra_map_injective A R B; simp)
 #align is_integral_closure.equiv IsIntegralClosure.equiv
 -/
 
@@ -1419,24 +1383,19 @@ theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x
   generalize hS : (↑(p.map <| algebraMap A B).frange : Set B) = S
   have coeffs_mem : ∀ i, (p.map <| algebraMap A B).coeff i ∈ adjoin R S :=
     by
-    intro i
-    by_cases hi : (p.map <| algebraMap A B).coeff i = 0
-    · rw [hi]
-      exact Subalgebra.zero_mem _
+    intro i; by_cases hi : (p.map <| algebraMap A B).coeff i = 0
+    · rw [hi]; exact Subalgebra.zero_mem _
     rw [← hS]
     exact subset_adjoin (coeff_mem_frange _ _ hi)
   obtain ⟨q, hq⟩ :
-    ∃ q : (adjoin R S)[X], q.map (algebraMap (adjoin R S) B) = (p.map <| algebraMap A B) :=
-    by
-    rw [← Set.mem_range]
-    exact (Polynomial.mem_map_range _).2 fun i => ⟨⟨_, coeffs_mem i⟩, rfl⟩
+    ∃ q : (adjoin R S)[X], q.map (algebraMap (adjoin R S) B) = (p.map <| algebraMap A B) := by
+    rw [← Set.mem_range]; exact (Polynomial.mem_map_range _).2 fun i => ⟨⟨_, coeffs_mem i⟩, rfl⟩
   use q
   constructor
   · suffices h : (q.map (algebraMap (adjoin R S) B)).Monic
     · refine' monic_of_injective _ h
       exact Subtype.val_injective
-    · rw [hq]
-      exact pmonic.map _
+    · rw [hq]; exact pmonic.map _
   · convert hp using 1
     replace hq := congr_arg (eval x) hq
     convert hq using 1 <;> symm <;> apply eval_map

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

@@ -136,10 +136,7 @@ theorem isIntegral_of_noetherian (H : IsNoetherian R A) (x : A) : IsIntegral R x
 #align is_integral_of_noetherian isIntegral_of_noetherian
 
 /- warning: is_integral_of_submodule_noetherian -> isIntegral_of_submodule_noetherian is a dubious translation:
-lean 3 declaration is
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A _inst_2) _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) => LE.le.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (SetLike.instPartialOrder.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) => LE.le.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) S))) -> (forall (x : A), (Membership.mem.{u1, u1} A (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (SetLike.instMembership.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) x S) -> (IsIntegral.{u2, u1} R A _inst_1 _inst_2 _inst_4 x))
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_of_submodule_noetherian isIntegral_of_submodule_noetherianₓ'. -/
 theorem isIntegral_of_submodule_noetherian (S : Subalgebra R A) (H : IsNoetherian R S.toSubmodule)
     (x : A) (hx : x ∈ S) : IsIntegral R x :=
@@ -167,10 +164,7 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S]
 variable [Algebra R A] [Algebra R B] (f : R →+* S)
 
 /- warning: map_is_integral -> map_isIntegral is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {B : Type.{u3}} {C : Type.{u4}} {F : Type.{u5}} [_inst_7 : Ring.{u3} B] [_inst_8 : Ring.{u4} C] [_inst_9 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B _inst_7)] [_inst_10 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7)] [_inst_11 : Algebra.{u1, u4} R C (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} C _inst_8)] [_inst_12 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B _inst_7) _inst_9)))))] [_inst_13 : Algebra.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8)] [_inst_14 : IsScalarTower.{u1, u2, u4} R A C (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u4} A C (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (SMulWithZero.toSmulZeroClass.{u2, u4} A C (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (MulActionWithZero.toSMulWithZero.{u2, u4} A C (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (Module.toMulActionWithZero.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13))))) (SMulZeroClass.toHasSmul.{u1, u4} R C (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (SMulWithZero.toSmulZeroClass.{u1, u4} R C (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.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (MulActionWithZero.toSMulWithZero.{u1, u4} R C (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (Module.toMulActionWithZero.{u1, u4} R C (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} R C (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} C _inst_8) _inst_11)))))] {b : B} [_inst_15 : AlgHomClass.{u5, u2, u3, u4} F A B C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13] (f : F), (IsIntegral.{u1, u3} R B _inst_1 _inst_7 _inst_9 b) -> (IsIntegral.{u1, u4} R C _inst_1 _inst_8 _inst_11 (coeFn.{succ u5, max (succ u3) (succ u4)} F (fun (_x : F) => B -> C) (FunLike.hasCoeToFun.{succ u5, succ u3, succ u4} F B (fun (_x : B) => C) (SMulHomClass.toFunLike.{u5, u2, u3, u4} F A B C (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (DistribSMul.toSmulZeroClass.{u2, u3} A B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))))) (DistribMulAction.toDistribSMul.{u2, u3} A B (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))) (Module.toDistribMulAction.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10))))) (SMulZeroClass.toHasSmul.{u2, u4} A C (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (DistribSMul.toSmulZeroClass.{u2, u4} A C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribMulAction.toDistribSMul.{u2, u4} A C (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13))))) (DistribMulActionHomClass.toSmulHomClass.{u5, u2, u3, u4} F A B C (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u5, u2, u3, u4} F A B C (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))) (Module.toDistribMulAction.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (AlgHom.nonUnitalAlgHomClass.{u2, u3, u4, u5} A B C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13 F _inst_15))))) f b))
-but is expected to have type
-  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {B : Type.{u5}} {C : Type.{u4}} {F : Type.{u3}} [_inst_7 : Ring.{u5} B] [_inst_8 : Ring.{u4} C] [_inst_9 : Algebra.{u2, u5} R B (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u5} B _inst_7)] [_inst_10 : Algebra.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7)] [_inst_11 : Algebra.{u2, u4} R C (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u4} C _inst_8)] [_inst_12 : IsScalarTower.{u2, u1, u5} R A B (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.toSMul.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10) (Algebra.toSMul.{u2, u5} R B (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u5} B _inst_7) _inst_9)] [_inst_13 : Algebra.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8)] [_inst_14 : IsScalarTower.{u2, u1, u4} R A C (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.toSMul.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13) (Algebra.toSMul.{u2, u4} R C (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u4} C _inst_8) _inst_11)] {b : B} [_inst_15 : AlgHomClass.{u3, u1, u5, u4} F A B C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13] (f : F), (IsIntegral.{u2, u5} R B _inst_1 _inst_7 _inst_9 b) -> (IsIntegral.{u2, u4} R ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : B) => C) b) _inst_1 _inst_8 _inst_11 (FunLike.coe.{succ u3, succ u5, succ u4} F B (fun (_x : B) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : B) => C) _x) (SMulHomClass.toFunLike.{u3, u1, u5, u4} F A B C (SMulZeroClass.toSMul.{u1, u5} A B (AddMonoid.toZero.{u5} B (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u5} A B (AddMonoid.toAddZeroClass.{u5} B (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u5} A B (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10))))) (SMulZeroClass.toSMul.{u1, u4} A C (AddMonoid.toZero.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u4} A C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u4} A C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13))))) (DistribMulActionHomClass.toSMulHomClass.{u3, u1, u5, u4} F A B C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u3, u1, u5, u4} F A B C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u5, u4, u3} A B C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13 F _inst_15)))) f b))
+<too large>
 Case conversion may be inaccurate. Consider using '#align map_is_integral map_isIntegralₓ'. -/
 theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
     [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B} [AlgHomClass F A B C] (f : F)
@@ -183,10 +177,7 @@ theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra
 #align map_is_integral map_isIntegral
 
 /- warning: is_integral_map_of_comp_eq_of_is_integral -> isIntegral_map_of_comp_eq_of_isIntegral is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_map_of_comp_eq_of_is_integral isIntegral_map_of_comp_eq_of_isIntegralₓ'. -/
 theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R] [CommRing S]
     [CommRing T] [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
@@ -200,10 +191,7 @@ theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R]
 #align is_integral_map_of_comp_eq_of_is_integral isIntegral_map_of_comp_eq_of_isIntegral
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_alg_hom_iff isIntegral_algHom_iffₓ'. -/
 theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A →ₐ[R] B) (hf : Function.Injective f) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
@@ -216,10 +204,7 @@ theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Al
 #align is_integral_alg_hom_iff isIntegral_algHom_iff
 
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_inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) (AlgEquivClass.toAlgHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgEquiv.instAlgEquivClassAlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10)))))) f x)) (IsIntegral.{u1, u3} R A _inst_1 _inst_7 _inst_9 x)
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_alg_equiv isIntegral_algEquivₓ'. -/
 @[simp]
 theorem isIntegral_algEquiv {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
@@ -261,10 +246,7 @@ theorem isIntegral_ofSubring {x : A} (T : Subring R) (hx : IsIntegral T x) : IsI
 #align is_integral_of_subring isIntegral_ofSubring
 
 /- warning: is_integral.algebra_map -> IsIntegral.algebraMap is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_6 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_7 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_8 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6)))))] {x : A}, (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x) -> (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_6 (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7) x))
-but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u3}} {B : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u2} B] [_inst_5 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_6 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_7 : Algebra.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_8 : IsScalarTower.{u1, u3, u2} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_5) (Algebra.toSMul.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) (Algebra.toSMul.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)] {x : A}, (IsIntegral.{u1, u3} R A _inst_1 (CommRing.toRing.{u3} A _inst_2) _inst_5 x) -> (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_1 (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_3) _inst_6 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonUnitalNonAssocSemiring.toMul.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) x))
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral.algebra_map IsIntegral.algebraMapₓ'. -/
 theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A} (h : IsIntegral R x) :
     IsIntegral R (algebraMap A B x) :=
@@ -275,10 +257,7 @@ theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A} (h : I
 #align is_integral.algebra_map IsIntegral.algebraMap
 
 /- warning: is_integral_algebra_map_iff -> isIntegral_algebraMap_iff is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_6 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_7 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_8 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6)))))] {x : A}, (Function.Injective.{succ u2, succ u3} A B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7))) -> (Iff (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_6 (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7) x)) (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x))
-but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u3}} {B : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u2} B] [_inst_5 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_6 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_7 : Algebra.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_8 : IsScalarTower.{u1, u3, u2} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_5) (Algebra.toSMul.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) (Algebra.toSMul.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)] {x : A}, (Function.Injective.{succ u3, succ u2} A B (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonUnitalNonAssocSemiring.toMul.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7))) -> (Iff (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_1 (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_3) _inst_6 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonUnitalNonAssocSemiring.toMul.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) x)) (IsIntegral.{u1, u3} R A _inst_1 (CommRing.toRing.{u3} A _inst_2) _inst_5 x))
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_algebra_map_iff isIntegral_algebraMap_iffₓ'. -/
 theorem isIntegral_algebraMap_iff [Algebra A B] [IsScalarTower R A B] {x : A}
     (hAB : Function.Injective (algebraMap A B)) :
@@ -304,10 +283,7 @@ theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
 #align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finite
 
 /- warning: fg_adjoin_singleton_of_integral -> FG_adjoin_singleton_of_integral is a dubious translation:
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-but is expected to have type
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(Subalgebra.toSubmodule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.adjoin.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5 (Singleton.singleton.{u1, u1} A (Set.{u1} A) (Set.instSingletonSet.{u1} A) x))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align fg_adjoin_singleton_of_integral FG_adjoin_singleton_of_integralₓ'. -/
 theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
     (Algebra.adjoin R ({x} : Set A)).toSubmodule.FG :=
@@ -341,10 +317,7 @@ theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
 #align fg_adjoin_singleton_of_integral FG_adjoin_singleton_of_integral
 
 /- warning: fg_adjoin_of_finite -> FG_adjoin_of_finite is a dubious translation:
-lean 3 declaration is
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-but is expected to have type
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+<too large>
 Case conversion may be inaccurate. Consider using '#align fg_adjoin_of_finite FG_adjoin_of_finiteₓ'. -/
 theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
     (Algebra.adjoin R s).toSubmodule.FG :=
@@ -365,10 +338,7 @@ theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
 #align fg_adjoin_of_finite FG_adjoin_of_finite
 
 /- warning: is_noetherian_adjoin_finset -> isNoetherian_adjoin_finset is a dubious translation:
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+<too large>
 Case conversion may be inaccurate. Consider using '#align is_noetherian_adjoin_finset isNoetherian_adjoin_finsetₓ'. -/
 theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
     (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (↑s : Set A)) :=
@@ -376,10 +346,7 @@ theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
 #align is_noetherian_adjoin_finset isNoetherian_adjoin_finset
 
 /- warning: is_integral_of_mem_of_fg -> isIntegral_of_mem_of_FG is a dubious translation:
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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R 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(CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) S)) -> (forall (x : A), (Membership.mem.{u1, u1} A (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (SetLike.instMembership.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) x S) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x))
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_of_mem_of_fg isIntegral_of_mem_of_FGₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
@@ -495,10 +462,7 @@ theorem Module.End.isIntegral {M : Type _} [AddCommGroup M] [Module R M] [Module
 #align module.End.is_integral Module.End.isIntegral
 
 /- warning: is_integral_of_smul_mem_submodule -> isIntegral_of_smul_mem_submodule is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {M : Type.{u3}} [_inst_7 : AddCommGroup.{u3} M] [_inst_8 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)] [_inst_9 : Module.{u2, u3} A M (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)] [_inst_10 : IsScalarTower.{u1, u2, u3} R A M (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} A M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} A M (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} A M (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (Module.toMulActionWithZero.{u2, u3} A M (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9)))) (SMulZeroClass.toHasSmul.{u1, u3} R M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (SMulWithZero.toSmulZeroClass.{u1, u3} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (MulActionWithZero.toSMulWithZero.{u1, u3} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (Module.toMulActionWithZero.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8))))] [_inst_11 : NoZeroSMulDivisors.{u2, u3} A M (MulZeroClass.toHasZero.{u2} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (SubNegMonoid.toAddMonoid.{u3} M (AddGroup.toSubNegMonoid.{u3} M (AddCommGroup.toAddGroup.{u3} M _inst_7))))) (SMulZeroClass.toHasSmul.{u2, u3} A M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} A M (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} A M (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (Module.toMulActionWithZero.{u2, u3} A M (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9))))] (N : Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8), (Ne.{succ u3} (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) N (Bot.bot.{u3} (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (Submodule.hasBot.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8))) -> (Submodule.FG.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8 N) -> (forall (x : A), (forall (n : M), (Membership.Mem.{u3, u3} M (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) M (Submodule.setLike.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8)) n N) -> (Membership.Mem.{u3, u3} M (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) M (Submodule.setLike.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8)) (SMul.smul.{u2, u3} A M (SMulZeroClass.toHasSmul.{u2, u3} A M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} A M (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} A M (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (Module.toMulActionWithZero.{u2, u3} A M (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9)))) x n) N)) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x))
-but is expected to have type
-  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {M : Type.{u3}} [_inst_7 : AddCommGroup.{u3} M] [_inst_8 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)] [_inst_9 : Module.{u1, u3} A M (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)] [_inst_10 : IsScalarTower.{u2, u1, u3} R A M (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (SMulZeroClass.toSMul.{u1, u3} A M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u3} A M (CommMonoidWithZero.toZero.{u1} A (CommSemiring.toCommMonoidWithZero.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u3} A M (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (Module.toMulActionWithZero.{u1, u3} A M (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9)))) (SMulZeroClass.toSMul.{u2, u3} R M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (Module.toMulActionWithZero.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8))))] [_inst_11 : NoZeroSMulDivisors.{u1, u3} A M (CommMonoidWithZero.toZero.{u1} A (CommSemiring.toCommMonoidWithZero.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (SMulZeroClass.toSMul.{u1, u3} A M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u3} A M (CommMonoidWithZero.toZero.{u1} A (CommSemiring.toCommMonoidWithZero.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u3} A M (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (Module.toMulActionWithZero.{u1, u3} A M (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9))))] (N : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8), (Ne.{succ u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (Submodule.instBotSubmodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8))) -> (Submodule.FG.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8 N) -> (forall (x : A), (forall (n : M), (Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8)) n N) -> (Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8)) (HSMul.hSMul.{u1, u3, u3} A M M (instHSMul.{u1, u3} A M (SMulZeroClass.toSMul.{u1, u3} A M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u3} A M (CommMonoidWithZero.toZero.{u1} A (CommSemiring.toCommMonoidWithZero.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u3} A M (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (Module.toMulActionWithZero.{u1, u3} A M (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9))))) x n) N)) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x))
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_of_smul_mem_submodule isIntegral_of_smul_mem_submoduleₓ'. -/
 /-- Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
 and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. -/
@@ -837,10 +801,7 @@ def integralClosure : Subalgebra R A
 -/
 
 /- warning: mem_integral_closure_iff_mem_fg -> mem_integralClosure_iff_mem_FG is a dubious translation:
-lean 3 declaration is
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(Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)))))))) (Subalgebra.toSubmodule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) M)) (Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) r M)))
-but is expected to have type
-  forall (R : Type.{u1}) (A : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] {r : A}, Iff (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) r (integralClosure.{u1, u2} R A _inst_1 _inst_2 _inst_5)) (Exists.{succ u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (fun (M : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) => And (Submodule.FG.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5))))))) (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (fun (_x : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5))))))) (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) => LE.le.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)))) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A 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+<too large>
 Case conversion may be inaccurate. Consider using '#align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_FGₓ'. -/
 theorem mem_integralClosure_iff_mem_FG {r : A} :
     r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M :=
@@ -880,10 +841,7 @@ theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
 #align le_integral_closure_iff_is_integral le_integralClosure_iff_isIntegral
 
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 Case conversion may be inaccurate. Consider using '#align is_integral_sup isIntegral_supₓ'. -/
 theorem isIntegral_sup {S T : Subalgebra R A} :
     Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R S ∧ Algebra.IsIntegral R T := by
@@ -891,10 +849,7 @@ theorem isIntegral_sup {S T : Subalgebra R A} :
 #align is_integral_sup isIntegral_sup
 
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 Case conversion may be inaccurate. Consider using '#align integral_closure_map_alg_equiv integralClosure_map_algEquivₓ'. -/
 /-- Mapping an integral closure along an `alg_equiv` gives the integral closure. -/
 theorem integralClosure_map_algEquiv (f : A ≃ₐ[R] B) :
@@ -1163,10 +1118,7 @@ theorem normalizeScaleRoots_degree : (normalizeScaleRoots p).degree = p.degree :
 -/
 
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+<too large>
 Case conversion may be inaccurate. Consider using '#align normalize_scale_roots_eval₂_leading_coeff_mul normalizeScaleRoots_eval₂_leadingCoeff_mulₓ'. -/
 theorem normalizeScaleRoots_eval₂_leadingCoeff_mul (h : 1 ≤ p.natDegree) (f : R →+* S) (x : S) :
     (normalizeScaleRoots p).eval₂ f (f p.leadingCoeff * x) =
@@ -1221,10 +1173,7 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
 #align ring_hom.is_integral_elem_leading_coeff_mul RingHom.isIntegralElem_leadingCoeff_mul
 
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_inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (fun (_x : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) _x) (SMulHomClass.toFunLike.{max u1 u2, u2, u2, u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (SMulZeroClass.toSMul.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddMonoid.toZero.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddMonoid.toAddZeroClass.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) (Module.toDistribMulAction.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Algebra.toModule.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (SMulZeroClass.toSMul.{u2, u1} R S (AddMonoid.toZero.{u1} S (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))))) (DistribSMul.toSMulZeroClass.{u2, u1} R S (AddMonoid.toAddZeroClass.{u1} S (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))))) (DistribMulAction.toDistribSMul.{u2, u1} R S (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7))))) (DistribMulActionHomClass.toSMulHomClass.{max u1 u2, u2, u2, u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)))))) (Module.toDistribMulAction.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Algebra.toModule.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u1 u2, u2, u2, u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{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_4)))) (Module.toDistribMulAction.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Algebra.toModule.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u2, u2, u1, max u1 u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7 (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) (AlgHom.algHomClass.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7))))) (Polynomial.aeval.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7 x) p) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommMonoidWithZero.toZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommSemiring.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommRing.toCommSemiring.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) _inst_4)))))) -> (IsIntegral.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) _inst_7 (HSMul.hSMul.{u2, u1, u1} R S S (instHSMul.{u2, u1} R S (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7)) (Polynomial.leadingCoeff.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) p) x))
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_leading_coeff_smul isIntegral_leadingCoeff_smulₓ'. -/
 /-- Given a `p : R[X]` and a root `x : S`,
 then `p.leading_coeff • x : S` is integral over `R`. -/
@@ -1299,10 +1248,7 @@ theorem isIntegral_algebra [Algebra R A] [IsScalarTower R A B] : Algebra.IsInteg
 #align is_integral_closure.is_integral_algebra IsIntegralClosure.isIntegral_algebra
 
 /- warning: is_integral_closure.no_zero_smul_divisors -> IsIntegralClosure.noZeroSMulDivisors is a dubious translation:
-lean 3 declaration is
-  forall (R : Type.{u1}) {A : Type.{u2}} (B : Type.{u3}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_7))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] [_inst_9 : NoZeroSMulDivisors.{u1, u3} R B (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (MulZeroClass.toHasZero.{u3} B (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} B (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} B (NonAssocRing.toNonUnitalNonAssocRing.{u3} B (Ring.toNonAssocRing.{u3} B (CommRing.toRing.{u3} B _inst_3)))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))], NoZeroSMulDivisors.{u1, u2} R A (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (MulZeroClass.toHasZero.{u2} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_7)))))
-but is expected to have type
-  forall (R : Type.{u3}) {A : Type.{u2}} (B : Type.{u1}) [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u1} B] [_inst_4 : Algebra.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_5 : Algebra.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u3, u1} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u3, u2, u1} R A B (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_7) (Algebra.toSMul.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_5) (Algebra.toSMul.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_4)] [_inst_9 : NoZeroSMulDivisors.{u3, u1} R B (CommMonoidWithZero.toZero.{u3} R (CommSemiring.toCommMonoidWithZero.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (CommMonoidWithZero.toZero.{u1} B (CommSemiring.toCommMonoidWithZero.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))) (Algebra.toSMul.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_4)], NoZeroSMulDivisors.{u3, u2} R A (CommMonoidWithZero.toZero.{u3} R (CommSemiring.toCommMonoidWithZero.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (CommMonoidWithZero.toZero.{u2} A (CommSemiring.toCommMonoidWithZero.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_7)
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_closure.no_zero_smul_divisors IsIntegralClosure.noZeroSMulDivisorsₓ'. -/
 theorem noZeroSMulDivisors [Algebra R A] [IsScalarTower R A B] [NoZeroSMulDivisors R B] :
     NoZeroSMulDivisors R A :=
@@ -1380,10 +1326,7 @@ theorem mk'_mul (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
 #align is_integral_closure.mk'_mul IsIntegralClosure.mk'_mul
 
 /- warning: is_integral_closure.mk'_algebra_map -> IsIntegralClosure.mk'_algebraMap is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} (A : Type.{u2}) {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_7))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B 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(CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4) x)) (isIntegral_algebraMap.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_4 x)), Eq.{succ u2} A (IsIntegralClosure.mk'.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (RingHom.{u1, u3} R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u1, u3} R B (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => 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(CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))) (algebraMap.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_7) x)
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-  forall {R : Type.{u3}} (A : Type.{u2}) {B : Type.{u1}} [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u1} B] [_inst_4 : Algebra.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_5 : Algebra.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u3, u1} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u3, u2, u1} R A B (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_7) (Algebra.toSMul.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) 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(CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))))))) (algebraMap.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_4) x) h) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (RingHom.{u3, u2} R A (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => A) _x) (MulHomClass.toFunLike.{max u3 u2, u3, u2} (RingHom.{u3, u2} R A (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) R A (NonUnitalNonAssocSemiring.toMul.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} R A (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) R A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} R A (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) R A (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (RingHom.instRingHomClassRingHom.{u3, u2} R A (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))))) (algebraMap.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_7) x)
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_closure.mk'_algebra_map IsIntegralClosure.mk'_algebraMapₓ'. -/
 @[simp]
 theorem mk'_algebraMap [Algebra R A] [IsScalarTower R A B] (x : R)
@@ -1413,10 +1356,7 @@ noncomputable def lift : S →ₐ[R] A
 -/
 
 /- warning: is_integral_closure.algebra_map_lift -> IsIntegralClosure.algebraMap_lift is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} (A : Type.{u2}) (B : Type.{u3}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] {S : Type.{u4}} [_inst_7 : CommRing.{u4} S] [_inst_8 : Algebra.{u1, u4} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7))] [_inst_9 : Algebra.{u4, u3} S B (CommRing.toCommSemiring.{u4} S _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_10 : IsScalarTower.{u1, u4, u3} R S B (SMulZeroClass.toHasSmul.{u1, u4} R S (AddZeroClass.toHasZero.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)))))))) (SMulWithZero.toSmulZeroClass.{u1, u4} 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.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)))))))) (MulActionWithZero.toSMulWithZero.{u1, u4} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)))))))) (Module.toMulActionWithZero.{u1, u4} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7))))) (Algebra.toModule.{u1, u4} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) _inst_8))))) (SMulZeroClass.toHasSmul.{u4, u3} S B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u4, u3} S B (MulZeroClass.toHasZero.{u4} S (MulZeroOneClass.toMulZeroClass.{u4} S (MonoidWithZero.toMulZeroOneClass.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u4, u3} S B (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u4, u3} S B (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u4, u3} S B (CommRing.toCommSemiring.{u4} S _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_9))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] [_inst_11 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_12 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_11))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B 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u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] (h : Algebra.IsIntegral.{u1, u4} R S _inst_1 (CommRing.toRing.{u4} S _inst_7) _inst_8) (x : S), Eq.{succ u3} B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) (coeFn.{max (succ u4) (succ u2), max (succ u4) (succ u2)} (AlgHom.{u1, u4, u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8 _inst_11) (fun (_x : AlgHom.{u1, u4, u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8 _inst_11) => S -> A) ([anonymous].{u1, u4, u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8 _inst_11) (IsIntegralClosure.lift.{u1, u2, u3, u4} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 S _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 h) x)) (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (RingHom.{u4, u3} S B (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u4, u3} S B (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => S -> B) (RingHom.hasCoeToFun.{u4, u3} S B (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u4, u3} S B (CommRing.toCommSemiring.{u4} S _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_9) x)
-but is expected to have type
-  forall {R : Type.{u1}} (A : Type.{u3}) (B : Type.{u4}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u4} B] [_inst_4 : Algebra.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_5 : Algebra.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_6 : IsIntegralClosure.{u3, u1, u4} A R B _inst_1 (CommRing.toCommSemiring.{u3} A _inst_2) _inst_3 _inst_4 _inst_5] {S : Type.{u2}} [_inst_7 : CommRing.{u2} S] [_inst_8 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))] [_inst_9 : Algebra.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_10 : IsScalarTower.{u1, u2, u4} R S B (Algebra.toSMul.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8) (Algebra.toSMul.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_9) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] [_inst_11 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_12 : IsScalarTower.{u1, u3, u4} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11) (Algebra.toSMul.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] (h : Algebra.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_7) _inst_8) (x : S), Eq.{succ u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) S (fun (a : S) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : S) => A) a) (SMulHomClass.toFunLike.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (SMulZeroClass.toSMul.{u1, u2} R S (AddMonoid.toZero.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R S (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8))))) (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8)) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u2, u3, max u3 u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11 (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) (AlgHom.algHomClass.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11))))) (IsIntegralClosure.lift.{u1, u3, u4, u2} 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(CommRing.toCommSemiring.{u4} B _inst_3))))))) (algebraMap.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_9) x)
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_closure.algebra_map_lift IsIntegralClosure.algebraMap_liftₓ'. -/
 @[simp]
 theorem algebraMap_lift (x : S) : algebraMap A B (lift A B h x) = algebraMap S B x :=
@@ -1447,10 +1387,7 @@ noncomputable def equiv : A ≃ₐ[R] A' :=
 -/
 
 /- warning: is_integral_closure.algebra_map_equiv -> IsIntegralClosure.algebraMap_equiv is a dubious translation:
-lean 3 declaration is
-  forall (R : Type.{u1}) (A : Type.{u2}) (B : Type.{u3}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] (A' : Type.{u4}) [_inst_7 : CommRing.{u4} A'] [_inst_8 : Algebra.{u4, u3} A' B (CommRing.toCommSemiring.{u4} A' _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_9 : IsIntegralClosure.{u4, u1, u3} A' R B _inst_1 (CommRing.toCommSemiring.{u4} A' _inst_7) _inst_3 _inst_4 _inst_8] [_inst_10 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_11 : Algebra.{u1, u4} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7))] [_inst_12 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_10))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] [_inst_13 : IsScalarTower.{u1, u4, u3} R A' B (SMulZeroClass.toHasSmul.{u1, u4} R A' (AddZeroClass.toHasZero.{u4} A' (AddMonoid.toAddZeroClass.{u4} A' (AddCommMonoid.toAddMonoid.{u4} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)))))))) (SMulWithZero.toSmulZeroClass.{u1, u4} R A' (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.{u4} A' (AddMonoid.toAddZeroClass.{u4} A' (AddCommMonoid.toAddMonoid.{u4} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)))))))) (MulActionWithZero.toSMulWithZero.{u1, u4} R A' (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u4} A' (AddMonoid.toAddZeroClass.{u4} A' (AddCommMonoid.toAddMonoid.{u4} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)))))))) (Module.toMulActionWithZero.{u1, u4} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7))))) (Algebra.toModule.{u1, u4} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_11))))) (SMulZeroClass.toHasSmul.{u4, u3} A' B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u4, u3} A' B (MulZeroClass.toHasZero.{u4} A' (MulZeroOneClass.toMulZeroClass.{u4} A' (MonoidWithZero.toMulZeroOneClass.{u4} A' (Semiring.toMonoidWithZero.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u4, u3} A' B (Semiring.toMonoidWithZero.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u4, u3} A' B (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u4, u3} A' B (CommRing.toCommSemiring.{u4} A' _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_8))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] (x : A), Eq.{succ u3} B (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (RingHom.{u4, u3} A' B (Semiring.toNonAssocSemiring.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u4, u3} A' B (Semiring.toNonAssocSemiring.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A' -> B) (RingHom.hasCoeToFun.{u4, u3} A' B (Semiring.toNonAssocSemiring.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u4, u3} A' B (CommRing.toCommSemiring.{u4} A' _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_8) (coeFn.{max (succ u2) (succ u4), max (succ u2) (succ u4)} (AlgEquiv.{u1, u2, u4} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_10 _inst_11) (fun (_x : AlgEquiv.{u1, u2, u4} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_10 _inst_11) => A -> A') (AlgEquiv.hasCoeToFun.{u1, u2, u4} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_10 _inst_11) (IsIntegralClosure.equiv.{u1, u2, u3, u4} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 A' _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13) x)) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) x)
-but is expected to have type
-  forall (R : Type.{u1}) (A : Type.{u3}) (B : Type.{u4}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u4} B] [_inst_4 : Algebra.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_5 : Algebra.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_6 : IsIntegralClosure.{u3, u1, u4} A R B _inst_1 (CommRing.toCommSemiring.{u3} A _inst_2) _inst_3 _inst_4 _inst_5] (A' : Type.{u2}) [_inst_7 : CommRing.{u2} A'] [_inst_8 : Algebra.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_9 : IsIntegralClosure.{u2, u1, u4} A' R B _inst_1 (CommRing.toCommSemiring.{u2} A' _inst_7) _inst_3 _inst_4 _inst_8] [_inst_10 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_11 : Algebra.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))] [_inst_12 : IsScalarTower.{u1, u3, u4} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10) (Algebra.toSMul.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] [_inst_13 : IsScalarTower.{u1, u2, u4} R A' B (Algebra.toSMul.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11) (Algebra.toSMul.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_8) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] (x : A), Eq.{succ u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A') => B) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) A (fun (a : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => A') a) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10))))) (SMulZeroClass.toSMul.{u1, u2} R A' (AddMonoid.toZero.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R A' (AddMonoid.toAddZeroClass.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) (AlgEquivClass.toAlgHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.instAlgEquivClassAlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11)))))) (IsIntegralClosure.equiv.{u1, u3, u4, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 A' _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13) x)) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' (fun (_x : A') => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A') => B) _x) (MulHomClass.toFunLike.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (NonUnitalNonAssocSemiring.toMul.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (NonUnitalNonAssocSemiring.toMul.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))) (RingHom.instRingHomClassRingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))))) (algebraMap.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_8) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => A') _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10))))) (SMulZeroClass.toSMul.{u1, u2} R A' (AddMonoid.toZero.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R A' (AddMonoid.toAddZeroClass.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A 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+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_closure.algebra_map_equiv IsIntegralClosure.algebraMap_equivₓ'. -/
 @[simp]
 theorem algebraMap_equiv (x : A) : algebraMap A' B (equiv R A B A' x) = algebraMap A B x :=
@@ -1474,10 +1411,7 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S] [CommRing T]
 variable [Algebra A B] [Algebra R B] (f : R →+* S) (g : S →+* T)
 
 /- warning: is_integral_trans_aux -> isIntegral_trans_aux is a dubious translation:
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(CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (fun (_x : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) _x) (SMulHomClass.toFunLike.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B 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(CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (DistribSMul.toSMulZeroClass.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddMonoid.toAddZeroClass.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (DistribMulAction.toDistribSMul.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (SMulZeroClass.toSMul.{u3, u2} A B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (DistribSMul.toSMulZeroClass.{u3, u2} A B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (DistribMulAction.toDistribSMul.{u3, u2} A B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u3, u3, u2, max u2 u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6 (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) (AlgHom.algHomClass.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6))))) (Polynomial.aeval.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6 x) p) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommMonoidWithZero.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommSemiring.toCommMonoidWithZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommRing.toCommSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) _inst_3)))))) -> (IsIntegral.{u2, u2} (Subtype.{succ u2} B (fun (x : B) => Membership.mem.{u2, u2} B (Subalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) B (Subalgebra.instSetLikeSubalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7)) x (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p)))))) B (Subalgebra.toCommRing.{u1, u2} R B _inst_1 _inst_3 _inst_7 (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p))))) (CommRing.toRing.{u2} B _inst_3) (Subalgebra.toAlgebra.{u2, u1, u2} B R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u2} B _inst_3) (Ring.toSemiring.{u2} B (CommRing.toRing.{u2} B _inst_3)) _inst_7 (Algebra.id.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p))))) x)
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_trans_aux isIntegral_trans_auxₓ'. -/
 theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x p = 0) :
     IsIntegral (adjoin R (↑(p.map <| algebraMap A B).frange : Set B)) x :=
@@ -1580,10 +1514,7 @@ theorem isIntegral_of_surjective (h : Function.Surjective (algebraMap R A)) :
 #align is_integral_of_surjective isIntegral_of_surjective
 
 /- warning: is_integral_tower_bot_of_is_integral -> isIntegral_tower_bot_of_isIntegral is a dubious translation:
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(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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A 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(CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6) x)) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8 x))
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A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6))) -> (forall {x : A}, (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_1 (CommRing.toRing.{u2} ((fun 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) x)) -> (IsIntegral.{u1, u3} R A _inst_1 (CommRing.toRing.{u3} A _inst_2) _inst_8 x))
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_tower_bot_of_is_integral isIntegral_tower_bot_of_isIntegralₓ'. -/
 /-- If `R → A → B` is an algebra tower with `A → B` injective,
 then if the entire tower is an integral extension so is `R → A` -/
@@ -1613,10 +1544,7 @@ theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injective g)
 #align ring_hom.is_integral_tower_bot_of_is_integral RingHom.isIntegral_tower_bot_of_isIntegral
 
 /- warning: is_integral_tower_bot_of_is_integral_field -> isIntegral_tower_bot_of_isIntegral_field is a dubious translation:
-lean 3 declaration is
-  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_10 : CommRing.{u1} R] [_inst_11 : Field.{u2} A] [_inst_12 : CommRing.{u3} B] [_inst_13 : Nontrivial.{u3} B] [_inst_14 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_10) (Ring.toSemiring.{u2} A (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11)))] [_inst_15 : Algebra.{u2, u3} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12))] [_inst_16 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_10) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12))] [_inst_17 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11))))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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_10)))))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11))))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_10))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11))))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_10)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11)))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_10) (Ring.toSemiring.{u2} A (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11))) _inst_14))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11))))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12))))) (Algebra.toModule.{u2, u3} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)) _inst_15))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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_10)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_10))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_10)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_10) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)) _inst_16)))))] {x : A}, (IsIntegral.{u1, u3} R B _inst_10 (CommRing.toRing.{u3} B _inst_12) _inst_16 (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))) (algebraMap.{u2, u3} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)) _inst_15) x)) -> (IsIntegral.{u1, u2} R A _inst_10 (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11)) _inst_14 x)
-but is expected to have type
-  forall {R : Type.{u3}} {A : Type.{u2}} {B : Type.{u1}} [_inst_10 : CommRing.{u3} R] [_inst_11 : Field.{u2} A] [_inst_12 : CommRing.{u1} B] [_inst_13 : Nontrivial.{u1} B] [_inst_14 : Algebra.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_10) (DivisionSemiring.toSemiring.{u2} A (Semifield.toDivisionSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))] [_inst_15 : Algebra.{u2, u1} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12))] [_inst_16 : Algebra.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_10) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12))] [_inst_17 : IsScalarTower.{u3, u2, u1} R A B (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_10) (DivisionSemiring.toSemiring.{u2} A (Semifield.toDivisionSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11))) _inst_14) (Algebra.toSMul.{u2, u1} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)) _inst_15) (Algebra.toSMul.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_10) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)) _inst_16)] {x : A}, (IsIntegral.{u3, u1} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_10 (CommRing.toRing.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_12) _inst_16 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)))) A B (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))))) (NonUnitalNonAssocSemiring.toMul.{u1} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} B (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12))))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} B (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)))) A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12))) (RingHom.instRingHomClassRingHom.{u2, u1} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12))))))) (algebraMap.{u2, u1} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)) _inst_15) x)) -> (IsIntegral.{u3, u2} R A _inst_10 (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11)) _inst_14 x)
+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_tower_bot_of_is_integral_field isIntegral_tower_bot_of_isIntegral_fieldₓ'. -/
 theorem isIntegral_tower_bot_of_isIntegral_field {R A B : Type _} [CommRing R] [Field A]
     [CommRing B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]
@@ -1689,10 +1617,7 @@ theorem isIntegral_quotient_of_isIntegral {I : Ideal A} (hRA : Algebra.IsIntegra
 #align is_integral_quotient_of_is_integral isIntegral_quotient_of_isIntegral
 
 /- warning: is_integral_quotient_map_iff -> isIntegral_quotientMap_iff is a dubious translation:
-lean 3 declaration is
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+<too large>
 Case conversion may be inaccurate. Consider using '#align is_integral_quotient_map_iff isIntegral_quotientMap_iffₓ'. -/
 theorem isIntegral_quotientMap_iff {I : Ideal S} :
     (Ideal.quotientMap I f le_rfl).IsIntegral ↔
@@ -1789,10 +1714,7 @@ theorem Algebra.IsIntegral.isField_iff_isField {R S : Type _} [CommRing R] [Nont
 end Algebra
 
 /- warning: integral_closure_idem -> integralClosure_idem is a dubious translation:
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(CompleteLattice.toBot.{u1} (Subalgebra.{u1, u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) A (CommRing.toCommSemiring.{u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (Subalgebra.toCommRing.{u2, u1} R A _inst_1 _inst_2 _inst_3 (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Subalgebra.toAlgebra.{u1, u2, u1} A R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 (Algebra.id.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (Algebra.instCompleteLatticeSubalgebra.{u1, u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) A (CommRing.toCommSemiring.{u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (Subalgebra.toCommRing.{u2, u1} R A _inst_1 _inst_2 _inst_3 (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Subalgebra.toAlgebra.{u1, u2, u1} A R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 (Algebra.id.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3)))))
+<too large>
 Case conversion may be inaccurate. Consider using '#align integral_closure_idem integralClosure_idemₓ'. -/
 theorem integralClosure_idem {R : Type _} {A : Type _} [CommRing R] [CommRing A] [Algebra R A] :
     integralClosure (integralClosure R A : Set A) A = ⊥ :=

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

@@ -170,7 +170,7 @@ variable [Algebra R A] [Algebra R B] (f : R →+* S)
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {B : Type.{u3}} {C : Type.{u4}} {F : Type.{u5}} [_inst_7 : Ring.{u3} B] [_inst_8 : Ring.{u4} C] [_inst_9 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B _inst_7)] [_inst_10 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7)] [_inst_11 : Algebra.{u1, u4} R C (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} C _inst_8)] [_inst_12 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B _inst_7) _inst_9)))))] [_inst_13 : Algebra.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8)] [_inst_14 : IsScalarTower.{u1, u2, u4} R A C (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u4} A C (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (SMulWithZero.toSmulZeroClass.{u2, u4} A C (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (MulActionWithZero.toSMulWithZero.{u2, u4} A C (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (Module.toMulActionWithZero.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13))))) (SMulZeroClass.toHasSmul.{u1, u4} R C (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (SMulWithZero.toSmulZeroClass.{u1, u4} R C (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.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (MulActionWithZero.toSMulWithZero.{u1, u4} R C (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (Module.toMulActionWithZero.{u1, u4} R C (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} R C (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} C _inst_8) _inst_11)))))] {b : B} [_inst_15 : AlgHomClass.{u5, u2, u3, u4} F A B C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13] (f : F), (IsIntegral.{u1, u3} R B _inst_1 _inst_7 _inst_9 b) -> (IsIntegral.{u1, u4} R C _inst_1 _inst_8 _inst_11 (coeFn.{succ u5, max (succ u3) (succ u4)} F (fun (_x : F) => B -> C) (FunLike.hasCoeToFun.{succ u5, succ u3, succ u4} F B (fun (_x : B) => C) (SMulHomClass.toFunLike.{u5, u2, u3, u4} F A B C (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (DistribSMul.toSmulZeroClass.{u2, u3} A B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))))) (DistribMulAction.toDistribSMul.{u2, u3} A B (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))) (Module.toDistribMulAction.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10))))) (SMulZeroClass.toHasSmul.{u2, u4} A C (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (DistribSMul.toSmulZeroClass.{u2, u4} A C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribMulAction.toDistribSMul.{u2, u4} A C (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13))))) (DistribMulActionHomClass.toSmulHomClass.{u5, u2, u3, u4} F A B C (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u5, u2, u3, u4} F A B C (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))) (Module.toDistribMulAction.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (AlgHom.nonUnitalAlgHomClass.{u2, u3, u4, u5} A B C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13 F _inst_15))))) f b))
 but is expected to have type
-  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {B : Type.{u5}} {C : Type.{u4}} {F : Type.{u3}} [_inst_7 : Ring.{u5} B] [_inst_8 : Ring.{u4} C] [_inst_9 : Algebra.{u2, u5} R B (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u5} B _inst_7)] [_inst_10 : Algebra.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7)] [_inst_11 : Algebra.{u2, u4} R C (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u4} C _inst_8)] [_inst_12 : IsScalarTower.{u2, u1, u5} R A B (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.toSMul.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10) (Algebra.toSMul.{u2, u5} R B (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u5} B _inst_7) _inst_9)] [_inst_13 : Algebra.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8)] [_inst_14 : IsScalarTower.{u2, u1, u4} R A C (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.toSMul.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13) (Algebra.toSMul.{u2, u4} R C (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u4} C _inst_8) _inst_11)] {b : B} [_inst_15 : AlgHomClass.{u3, u1, u5, u4} F A B C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13] (f : F), (IsIntegral.{u2, u5} R B _inst_1 _inst_7 _inst_9 b) -> (IsIntegral.{u2, u4} R ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : B) => C) b) _inst_1 _inst_8 _inst_11 (FunLike.coe.{succ u3, succ u5, succ u4} F B (fun (_x : B) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : B) => C) _x) (SMulHomClass.toFunLike.{u3, u1, u5, u4} F A B C (SMulZeroClass.toSMul.{u1, u5} A B (AddMonoid.toZero.{u5} B (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u5} A B (AddMonoid.toAddZeroClass.{u5} B (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u5} A B (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10))))) (SMulZeroClass.toSMul.{u1, u4} A C (AddMonoid.toZero.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u4} A C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u4} A C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13))))) (DistribMulActionHomClass.toSMulHomClass.{u3, u1, u5, u4} F A B C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u3, u1, u5, u4} F A B C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u5, u4, u3} A B C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13 F _inst_15)))) f b))
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {B : Type.{u5}} {C : Type.{u4}} {F : Type.{u3}} [_inst_7 : Ring.{u5} B] [_inst_8 : Ring.{u4} C] [_inst_9 : Algebra.{u2, u5} R B (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u5} B _inst_7)] [_inst_10 : Algebra.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7)] [_inst_11 : Algebra.{u2, u4} R C (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u4} C _inst_8)] [_inst_12 : IsScalarTower.{u2, u1, u5} R A B (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.toSMul.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10) (Algebra.toSMul.{u2, u5} R B (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u5} B _inst_7) _inst_9)] [_inst_13 : Algebra.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8)] [_inst_14 : IsScalarTower.{u2, u1, u4} R A C (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.toSMul.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13) (Algebra.toSMul.{u2, u4} R C (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u4} C _inst_8) _inst_11)] {b : B} [_inst_15 : AlgHomClass.{u3, u1, u5, u4} F A B C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13] (f : F), (IsIntegral.{u2, u5} R B _inst_1 _inst_7 _inst_9 b) -> (IsIntegral.{u2, u4} R ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : B) => C) b) _inst_1 _inst_8 _inst_11 (FunLike.coe.{succ u3, succ u5, succ u4} F B (fun (_x : B) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : B) => C) _x) (SMulHomClass.toFunLike.{u3, u1, u5, u4} F A B C (SMulZeroClass.toSMul.{u1, u5} A B (AddMonoid.toZero.{u5} B (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u5} A B (AddMonoid.toAddZeroClass.{u5} B (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u5} A B (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10))))) (SMulZeroClass.toSMul.{u1, u4} A C (AddMonoid.toZero.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u4} A C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u4} A C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13))))) (DistribMulActionHomClass.toSMulHomClass.{u3, u1, u5, u4} F A B C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u3, u1, u5, u4} F A B C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u5, u4, u3} A B C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13 F _inst_15)))) f b))
 Case conversion may be inaccurate. Consider using '#align map_is_integral map_isIntegralₓ'. -/
 theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
     [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B} [AlgHomClass F A B C] (f : F)
@@ -203,7 +203,7 @@ theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R]
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {A : Type.{u2}} {B : Type.{u3}} [_inst_7 : Ring.{u2} A] [_inst_8 : Ring.{u3} B] [_inst_9 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7)] [_inst_10 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B _inst_8)] (f : AlgHom.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10), (Function.Injective.{succ u2, succ u3} A B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (AlgHom.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) (fun (_x : AlgHom.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) => A -> B) ([anonymous].{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) f)) -> (forall {x : A}, Iff (IsIntegral.{u1, u3} R B _inst_1 _inst_8 _inst_10 (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (AlgHom.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) (fun (_x : AlgHom.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) => A -> B) ([anonymous].{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) f x)) (IsIntegral.{u1, u2} R A _inst_1 _inst_7 _inst_9 x))
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {A : Type.{u3}} {B : Type.{u2}} [_inst_7 : Ring.{u3} A] [_inst_8 : Ring.{u2} B] [_inst_9 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7)] [_inst_10 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8)] (f : AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10), (Function.Injective.{succ u3, succ u2} A B (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9))))) (SMulZeroClass.toSMul.{u1, u2} R B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u2} R B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u2} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) (AlgHom.algHomClass.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10))))) f)) -> (forall {x : A}, Iff (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) x) _inst_1 _inst_8 _inst_10 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9))))) (SMulZeroClass.toSMul.{u1, u2} R B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u2} R B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u2} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) (AlgHom.algHomClass.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10))))) f x)) (IsIntegral.{u1, u3} R A _inst_1 _inst_7 _inst_9 x))
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {A : Type.{u3}} {B : Type.{u2}} [_inst_7 : Ring.{u3} A] [_inst_8 : Ring.{u2} B] [_inst_9 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7)] [_inst_10 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8)] (f : AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10), (Function.Injective.{succ u3, succ u2} A B (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => B) _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9))))) (SMulZeroClass.toSMul.{u1, u2} R B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u2} R B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u2} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) (AlgHom.algHomClass.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10))))) f)) -> (forall {x : A}, Iff (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => B) x) _inst_1 _inst_8 _inst_10 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => B) _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9))))) (SMulZeroClass.toSMul.{u1, u2} R B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u2} R B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u2} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) (AlgHom.algHomClass.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10))))) f x)) (IsIntegral.{u1, u3} R A _inst_1 _inst_7 _inst_9 x))
 Case conversion may be inaccurate. Consider using '#align is_integral_alg_hom_iff isIntegral_algHom_iffₓ'. -/
 theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A →ₐ[R] B) (hf : Function.Injective f) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
@@ -219,7 +219,7 @@ theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Al
 lean 3 declaration is
   forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {A : Type.{u2}} {B : Type.{u3}} [_inst_7 : Ring.{u2} A] [_inst_8 : Ring.{u3} B] [_inst_9 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7)] [_inst_10 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B _inst_8)] (f : AlgEquiv.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) {x : A}, Iff (IsIntegral.{u1, u3} R B _inst_1 _inst_8 _inst_10 (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (AlgEquiv.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) (fun (_x : AlgEquiv.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) => A -> B) (AlgEquiv.hasCoeToFun.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) f x)) (IsIntegral.{u1, u2} R A _inst_1 _inst_7 _inst_9 x)
 but is expected to have type
-  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {A : Type.{u3}} {B : Type.{u2}} [_inst_7 : Ring.{u3} A] [_inst_8 : Ring.{u2} B] [_inst_9 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7)] [_inst_10 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8)] (f : AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) {x : A}, Iff (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) x) _inst_1 _inst_8 _inst_10 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9))))) (SMulZeroClass.toSMul.{u1, u2} R B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u2} R B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u2} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) (AlgEquivClass.toAlgHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgEquiv.instAlgEquivClassAlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10)))))) f x)) (IsIntegral.{u1, u3} R A _inst_1 _inst_7 _inst_9 x)
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {A : Type.{u3}} {B : Type.{u2}} [_inst_7 : Ring.{u3} A] [_inst_8 : Ring.{u2} B] [_inst_9 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7)] [_inst_10 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8)] (f : AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) {x : A}, Iff (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => B) x) _inst_1 _inst_8 _inst_10 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => B) _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9))))) (SMulZeroClass.toSMul.{u1, u2} R B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u2} R B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u2} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) (AlgEquivClass.toAlgHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgEquiv.instAlgEquivClassAlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10)))))) f x)) (IsIntegral.{u1, u3} R A _inst_1 _inst_7 _inst_9 x)
 Case conversion may be inaccurate. Consider using '#align is_integral_alg_equiv isIntegral_algEquivₓ'. -/
 @[simp]
 theorem isIntegral_algEquiv {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
@@ -1224,7 +1224,7 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
 lean 3 declaration is
   forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] (p : Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (x : S) [_inst_7 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4))], (Eq.{succ u2} S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (AlgHom.{u1, u1, u2} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) S (CommRing.toCommSemiring.{u1} R _inst_1) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) (Polynomial.algebraOfAlgebra.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _inst_7) (fun (_x : AlgHom.{u1, u1, u2} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) S (CommRing.toCommSemiring.{u1} R _inst_1) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) (Polynomial.algebraOfAlgebra.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _inst_7) => (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) -> S) ([anonymous].{u1, u1, u2} R (Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) S (CommRing.toCommSemiring.{u1} R _inst_1) (Polynomial.semiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) (Polynomial.algebraOfAlgebra.{u1, u1} R R (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Algebra.id.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) _inst_7) (Polynomial.aeval.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) _inst_7 x) p) (OfNat.ofNat.{u2} S 0 (OfNat.mk.{u2} S 0 (Zero.zero.{u2} S (MulZeroClass.toHasZero.{u2} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))))))))) -> (IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) _inst_7 (SMul.smul.{u1, u2} R S (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_4)))))))) (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_4)))))))) (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_4)))))))) (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_4))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) _inst_7))))) (Polynomial.leadingCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) x))
 but is expected to have type
-  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_4 : CommRing.{u1} S] (p : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (x : S) [_inst_7 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))], (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (fun (_x : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) _x) (SMulHomClass.toFunLike.{max u1 u2, u2, u2, u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (SMulZeroClass.toSMul.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddMonoid.toZero.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddMonoid.toAddZeroClass.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R 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(CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7))))) (DistribMulActionHomClass.toSMulHomClass.{max u1 u2, u2, u2, u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)))))) (Module.toDistribMulAction.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Algebra.toModule.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u1 u2, u2, u2, u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{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_4)))) (Module.toDistribMulAction.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Algebra.toModule.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u2, u2, u1, max u1 u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7 (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) (AlgHom.algHomClass.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7))))) (Polynomial.aeval.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7 x) p) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommMonoidWithZero.toZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommSemiring.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommRing.toCommSemiring.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) _inst_4)))))) -> (IsIntegral.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) _inst_7 (HSMul.hSMul.{u2, u1, u1} R S S (instHSMul.{u2, u1} R S (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7)) (Polynomial.leadingCoeff.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) p) x))
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_4 : CommRing.{u1} S] (p : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (x : S) [_inst_7 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))], (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (FunLike.coe.{max (succ u1) (succ u2), succ u2, succ u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (fun (_x : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) _x) (SMulHomClass.toFunLike.{max u1 u2, u2, u2, u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (SMulZeroClass.toSMul.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddMonoid.toZero.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (DistribSMul.toSMulZeroClass.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddMonoid.toAddZeroClass.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (DistribMulAction.toDistribSMul.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) (Module.toDistribMulAction.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Algebra.toModule.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))))) (SMulZeroClass.toSMul.{u2, u1} R S (AddMonoid.toZero.{u1} S (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))))) (DistribSMul.toSMulZeroClass.{u2, u1} R S (AddMonoid.toAddZeroClass.{u1} S (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))))) (DistribMulAction.toDistribSMul.{u2, u1} R S (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7))))) (DistribMulActionHomClass.toSMulHomClass.{max u1 u2, u2, u2, u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))))) (AddCommMonoid.toAddMonoid.{u1} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)))))) (Module.toDistribMulAction.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Algebra.toModule.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u1 u2, u2, u2, u1} (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{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_4)))) (Module.toDistribMulAction.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))))) (Algebra.toModule.{u2, u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))))) (Module.toDistribMulAction.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))))) (Algebra.toModule.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u2, u2, u1, max u1 u2} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7 (AlgHom.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7) (AlgHom.algHomClass.{u2, u2, u1} R (Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7))))) (Polynomial.aeval.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7 x) p) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommMonoidWithZero.toZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommSemiring.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommRing.toCommSemiring.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) _inst_4)))))) -> (IsIntegral.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) _inst_7 (HSMul.hSMul.{u2, u1, u1} R S S (instHSMul.{u2, u1} R S (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7)) (Polynomial.leadingCoeff.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) p) x))
 Case conversion may be inaccurate. Consider using '#align is_integral_leading_coeff_smul isIntegral_leadingCoeff_smulₓ'. -/
 /-- Given a `p : R[X]` and a root `x : S`,
 then `p.leading_coeff • x : S` is integral over `R`. -/
@@ -1416,7 +1416,7 @@ noncomputable def lift : S →ₐ[R] A
 lean 3 declaration is
   forall {R : Type.{u1}} (A : Type.{u2}) (B : Type.{u3}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] {S : Type.{u4}} [_inst_7 : CommRing.{u4} S] [_inst_8 : Algebra.{u1, u4} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7))] [_inst_9 : Algebra.{u4, u3} S B (CommRing.toCommSemiring.{u4} S _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_10 : IsScalarTower.{u1, u4, u3} R S B (SMulZeroClass.toHasSmul.{u1, u4} R S (AddZeroClass.toHasZero.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)))))))) (SMulWithZero.toSmulZeroClass.{u1, u4} 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.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)))))))) (MulActionWithZero.toSMulWithZero.{u1, u4} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)))))))) (Module.toMulActionWithZero.{u1, u4} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7))))) (Algebra.toModule.{u1, u4} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) _inst_8))))) (SMulZeroClass.toHasSmul.{u4, u3} S B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u4, u3} S B (MulZeroClass.toHasZero.{u4} S (MulZeroOneClass.toMulZeroClass.{u4} S (MonoidWithZero.toMulZeroOneClass.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u4, u3} S B (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u4, u3} S B (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u4, u3} S B (CommRing.toCommSemiring.{u4} S _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_9))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] [_inst_11 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_12 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_11))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] (h : Algebra.IsIntegral.{u1, u4} R S _inst_1 (CommRing.toRing.{u4} S _inst_7) _inst_8) (x : S), Eq.{succ u3} B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) (coeFn.{max (succ u4) (succ u2), max (succ u4) (succ u2)} (AlgHom.{u1, u4, u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8 _inst_11) (fun (_x : AlgHom.{u1, u4, u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8 _inst_11) => S -> A) ([anonymous].{u1, u4, u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8 _inst_11) (IsIntegralClosure.lift.{u1, u2, u3, u4} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 S _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 h) x)) (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (RingHom.{u4, u3} S B (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u4, u3} S B (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => S -> B) (RingHom.hasCoeToFun.{u4, u3} S B (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u4, u3} S B (CommRing.toCommSemiring.{u4} S _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_9) x)
 but is expected to have type
-  forall {R : Type.{u1}} (A : Type.{u3}) (B : Type.{u4}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u4} B] [_inst_4 : Algebra.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_5 : Algebra.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_6 : IsIntegralClosure.{u3, u1, u4} A R B _inst_1 (CommRing.toCommSemiring.{u3} A _inst_2) _inst_3 _inst_4 _inst_5] {S : Type.{u2}} [_inst_7 : CommRing.{u2} S] [_inst_8 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))] [_inst_9 : Algebra.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_10 : IsScalarTower.{u1, u2, u4} R S B (Algebra.toSMul.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8) (Algebra.toSMul.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_9) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] [_inst_11 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_12 : IsScalarTower.{u1, u3, u4} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11) (Algebra.toSMul.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] (h : Algebra.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_7) _inst_8) (x : S), Eq.{succ u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) S (fun (a : S) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : S) => A) a) (SMulHomClass.toFunLike.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (SMulZeroClass.toSMul.{u1, u2} R S (AddMonoid.toZero.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R S (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8))))) (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8)) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{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_7)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8)) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u2, u3, max u3 u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11 (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) (AlgHom.algHomClass.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11))))) (IsIntegralClosure.lift.{u1, u3, u4, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 S _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 h) x)) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u4, u3, u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonUnitalNonAssocSemiring.toMul.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u4, u3, u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u4, u3, u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))))) (algebraMap.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) S (fun (_x : S) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : S) => A) _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (SMulZeroClass.toSMul.{u1, u2} R S (AddMonoid.toZero.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R S (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8))))) (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8)) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{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_7)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8)) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u2, u3, max u3 u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11 (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) (AlgHom.algHomClass.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11))))) (IsIntegralClosure.lift.{u1, u3, u4, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 S _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 h) x)) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (RingHom.{u2, u4} S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) S (fun (_x : S) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : S) => B) _x) (MulHomClass.toFunLike.{max u4 u2, u2, u4} (RingHom.{u2, u4} S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) S B (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (NonUnitalNonAssocSemiring.toMul.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) S B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))) (RingHom.instRingHomClassRingHom.{u2, u4} S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))))) (algebraMap.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_9) x)
+  forall {R : Type.{u1}} (A : Type.{u3}) (B : Type.{u4}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u4} B] [_inst_4 : Algebra.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_5 : Algebra.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_6 : IsIntegralClosure.{u3, u1, u4} A R B _inst_1 (CommRing.toCommSemiring.{u3} A _inst_2) _inst_3 _inst_4 _inst_5] {S : Type.{u2}} [_inst_7 : CommRing.{u2} S] [_inst_8 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))] [_inst_9 : Algebra.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_10 : IsScalarTower.{u1, u2, u4} R S B (Algebra.toSMul.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8) (Algebra.toSMul.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_9) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] [_inst_11 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_12 : IsScalarTower.{u1, u3, u4} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11) (Algebra.toSMul.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] (h : Algebra.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_7) _inst_8) (x : S), Eq.{succ u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) S (fun (a : S) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : S) => A) a) (SMulHomClass.toFunLike.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (SMulZeroClass.toSMul.{u1, u2} R S (AddMonoid.toZero.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R S (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} 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(AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u2, u3, max u3 u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11 (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) (AlgHom.algHomClass.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11))))) (IsIntegralClosure.lift.{u1, u3, u4, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 S _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 h) x)) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u4, u3, u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) 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(Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8)) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{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_7)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8)) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u2, u3, max u3 u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11 (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) (AlgHom.algHomClass.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11))))) (IsIntegralClosure.lift.{u1, u3, u4, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 S _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 h) x)) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (RingHom.{u2, u4} S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) S (fun (_x : S) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : S) => B) _x) (MulHomClass.toFunLike.{max u4 u2, u2, u4} (RingHom.{u2, u4} S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) S B (NonUnitalNonAssocSemiring.toMul.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (NonUnitalNonAssocSemiring.toMul.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) S B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))) (RingHom.instRingHomClassRingHom.{u2, u4} S B (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))))) (algebraMap.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_9) x)
 Case conversion may be inaccurate. Consider using '#align is_integral_closure.algebra_map_lift IsIntegralClosure.algebraMap_liftₓ'. -/
 @[simp]
 theorem algebraMap_lift (x : S) : algebraMap A B (lift A B h x) = algebraMap S B x :=
@@ -1450,7 +1450,7 @@ noncomputable def equiv : A ≃ₐ[R] A' :=
 lean 3 declaration is
   forall (R : Type.{u1}) (A : Type.{u2}) (B : Type.{u3}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] (A' : Type.{u4}) [_inst_7 : CommRing.{u4} A'] [_inst_8 : Algebra.{u4, u3} A' B (CommRing.toCommSemiring.{u4} A' _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_9 : IsIntegralClosure.{u4, u1, u3} A' R B _inst_1 (CommRing.toCommSemiring.{u4} A' _inst_7) _inst_3 _inst_4 _inst_8] [_inst_10 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_11 : Algebra.{u1, u4} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7))] [_inst_12 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_10))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] [_inst_13 : IsScalarTower.{u1, u4, u3} R A' B (SMulZeroClass.toHasSmul.{u1, u4} R A' (AddZeroClass.toHasZero.{u4} A' (AddMonoid.toAddZeroClass.{u4} A' (AddCommMonoid.toAddMonoid.{u4} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)))))))) (SMulWithZero.toSmulZeroClass.{u1, u4} R A' (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.{u4} A' (AddMonoid.toAddZeroClass.{u4} A' (AddCommMonoid.toAddMonoid.{u4} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)))))))) (MulActionWithZero.toSMulWithZero.{u1, u4} R A' (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u4} A' (AddMonoid.toAddZeroClass.{u4} A' (AddCommMonoid.toAddMonoid.{u4} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)))))))) (Module.toMulActionWithZero.{u1, u4} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7))))) (Algebra.toModule.{u1, u4} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_11))))) (SMulZeroClass.toHasSmul.{u4, u3} A' B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u4, u3} A' B (MulZeroClass.toHasZero.{u4} A' (MulZeroOneClass.toMulZeroClass.{u4} A' (MonoidWithZero.toMulZeroOneClass.{u4} A' (Semiring.toMonoidWithZero.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u4, u3} A' B (Semiring.toMonoidWithZero.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u4, u3} A' B (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u4, u3} A' B (CommRing.toCommSemiring.{u4} A' _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_8))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] (x : A), Eq.{succ u3} B (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (RingHom.{u4, u3} A' B (Semiring.toNonAssocSemiring.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u4, u3} A' B (Semiring.toNonAssocSemiring.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A' -> B) (RingHom.hasCoeToFun.{u4, u3} A' B (Semiring.toNonAssocSemiring.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u4, u3} A' B (CommRing.toCommSemiring.{u4} A' _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_8) (coeFn.{max (succ u2) (succ u4), max (succ u2) (succ u4)} (AlgEquiv.{u1, u2, u4} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_10 _inst_11) (fun (_x : AlgEquiv.{u1, u2, u4} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_10 _inst_11) => A -> A') (AlgEquiv.hasCoeToFun.{u1, u2, u4} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_10 _inst_11) (IsIntegralClosure.equiv.{u1, u2, u3, u4} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 A' _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13) x)) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) x)
 but is expected to have type
-  forall (R : Type.{u1}) (A : Type.{u3}) (B : Type.{u4}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u4} B] [_inst_4 : Algebra.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_5 : Algebra.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_6 : IsIntegralClosure.{u3, u1, u4} A R B _inst_1 (CommRing.toCommSemiring.{u3} A _inst_2) _inst_3 _inst_4 _inst_5] (A' : Type.{u2}) [_inst_7 : CommRing.{u2} A'] [_inst_8 : Algebra.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_9 : IsIntegralClosure.{u2, u1, u4} A' R B _inst_1 (CommRing.toCommSemiring.{u2} A' _inst_7) _inst_3 _inst_4 _inst_8] [_inst_10 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_11 : Algebra.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))] [_inst_12 : IsScalarTower.{u1, u3, u4} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10) (Algebra.toSMul.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] [_inst_13 : IsScalarTower.{u1, u2, u4} R A' B (Algebra.toSMul.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11) (Algebra.toSMul.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_8) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] (x : A), Eq.{succ u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A') => B) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) A (fun (a : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => A') a) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10))))) (SMulZeroClass.toSMul.{u1, u2} R A' (AddMonoid.toZero.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R A' (AddMonoid.toAddZeroClass.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) (AlgEquivClass.toAlgHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.instAlgEquivClassAlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11)))))) (IsIntegralClosure.equiv.{u1, u3, u4, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 A' _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13) x)) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' (fun (_x : A') => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A') => B) _x) (MulHomClass.toFunLike.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (NonUnitalNonAssocSemiring.toMul.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (NonUnitalNonAssocSemiring.toMul.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))) (RingHom.instRingHomClassRingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))))) (algebraMap.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_8) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => A') _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10))))) (SMulZeroClass.toSMul.{u1, u2} R A' (AddMonoid.toZero.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R A' (AddMonoid.toAddZeroClass.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) (AlgEquivClass.toAlgHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.instAlgEquivClassAlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11)))))) (IsIntegralClosure.equiv.{u1, u3, u4, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 A' _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13) x)) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u4, u3, u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonUnitalNonAssocSemiring.toMul.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u4, u3, u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u4, u3, u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))))) (algebraMap.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) x)
+  forall (R : Type.{u1}) (A : Type.{u3}) (B : Type.{u4}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u4} B] [_inst_4 : Algebra.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_5 : Algebra.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_6 : IsIntegralClosure.{u3, u1, u4} A R B _inst_1 (CommRing.toCommSemiring.{u3} A _inst_2) _inst_3 _inst_4 _inst_5] (A' : Type.{u2}) [_inst_7 : CommRing.{u2} A'] [_inst_8 : Algebra.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_9 : IsIntegralClosure.{u2, u1, u4} A' R B _inst_1 (CommRing.toCommSemiring.{u2} A' _inst_7) _inst_3 _inst_4 _inst_8] [_inst_10 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_11 : Algebra.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))] [_inst_12 : IsScalarTower.{u1, u3, u4} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10) (Algebra.toSMul.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] [_inst_13 : IsScalarTower.{u1, u2, u4} R A' B (Algebra.toSMul.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11) (Algebra.toSMul.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_8) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] (x : A), Eq.{succ u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A') => B) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) A (fun (a : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => A') a) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10))))) (SMulZeroClass.toSMul.{u1, u2} R A' (AddMonoid.toZero.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R A' (AddMonoid.toAddZeroClass.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) (AlgEquivClass.toAlgHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.instAlgEquivClassAlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11)))))) (IsIntegralClosure.equiv.{u1, u3, u4, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 A' _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13) x)) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' (fun (_x : A') => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A') => B) _x) (MulHomClass.toFunLike.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (NonUnitalNonAssocSemiring.toMul.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (NonUnitalNonAssocSemiring.toMul.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))) (RingHom.instRingHomClassRingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))))) (algebraMap.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_8) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : A) => A') _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A 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(CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R A' (AddMonoid.toAddZeroClass.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) (AlgEquivClass.toAlgHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.instAlgEquivClassAlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11)))))) (IsIntegralClosure.equiv.{u1, u3, u4, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 A' _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13) x)) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u4, u3, u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonUnitalNonAssocSemiring.toMul.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u4, u3, u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u4, u3, u4} (RingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u4} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))))) (algebraMap.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) x)
 Case conversion may be inaccurate. Consider using '#align is_integral_closure.algebra_map_equiv IsIntegralClosure.algebraMap_equivₓ'. -/
 @[simp]
 theorem algebraMap_equiv (x : A) : algebraMap A' B (equiv R A B A' x) = algebraMap A B x :=
@@ -1477,7 +1477,7 @@ variable [Algebra A B] [Algebra R B] (f : R →+* S) (g : S →+* T)
 lean 3 declaration is
   forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_6 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_7 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] (x : B) {p : Polynomial.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))}, (Polynomial.Monic.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) p) -> (Eq.{succ u3} B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (AlgHom.{u2, u2, u3} A (Polynomial.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) B (CommRing.toCommSemiring.{u2} A _inst_2) (Polynomial.semiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (Polynomial.algebraOfAlgebra.{u2, u2} A A (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Algebra.id.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) _inst_6) (fun (_x : AlgHom.{u2, u2, u3} A (Polynomial.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) B (CommRing.toCommSemiring.{u2} A _inst_2) (Polynomial.semiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (Polynomial.algebraOfAlgebra.{u2, u2} A A (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Algebra.id.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) _inst_6) => (Polynomial.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) -> B) ([anonymous].{u2, u2, u3} A (Polynomial.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) B (CommRing.toCommSemiring.{u2} A _inst_2) (Polynomial.semiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (Polynomial.algebraOfAlgebra.{u2, u2} A A (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Algebra.id.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) _inst_6) (Polynomial.aeval.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6 x) p) (OfNat.ofNat.{u3} B 0 (OfNat.mk.{u3} B 0 (Zero.zero.{u3} B (MulZeroClass.toHasZero.{u3} B (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} B (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} B (NonAssocRing.toNonUnitalNonAssocRing.{u3} B (Ring.toNonAssocRing.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))))) -> (IsIntegral.{u3, u3} (coeSort.{succ u3, succ (succ u3)} (Subalgebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Subalgebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7) B (Subalgebra.setLike.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7)) (Algebra.adjoin.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7 ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} B) (Set.{u3} B) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (Finset.Set.hasCoeT.{u3} B))) (Polynomial.frange.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (Polynomial.map.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6) p))))) B (Subalgebra.toCommRing.{u1, u3} R B _inst_1 _inst_3 _inst_7 (Algebra.adjoin.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7 ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} B) (Set.{u3} B) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (Finset.Set.hasCoeT.{u3} B))) (Polynomial.frange.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (Polynomial.map.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6) p))))) (CommRing.toRing.{u3} B _inst_3) (Subalgebra.toAlgebra.{u3, u1, u3} B R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u3} B _inst_3) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7 (Algebra.id.{u3} B (CommRing.toCommSemiring.{u3} B _inst_3)) (Algebra.adjoin.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7 ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} B) (Set.{u3} B) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (Finset.Set.hasCoeT.{u3} B))) (Polynomial.frange.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (Polynomial.map.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6) p))))) x)
 but is expected to have type
-  forall {R : Type.{u1}} {A : Type.{u3}} {B : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u2} B] [_inst_6 : Algebra.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_7 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] (x : B) {p : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))}, (Polynomial.Monic.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) p) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (fun (_x : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) _x) (SMulHomClass.toFunLike.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (SMulZeroClass.toSMul.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddMonoid.toZero.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (DistribSMul.toSMulZeroClass.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddMonoid.toAddZeroClass.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (DistribMulAction.toDistribSMul.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (SMulZeroClass.toSMul.{u3, u2} A B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (DistribSMul.toSMulZeroClass.{u3, u2} A B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (DistribMulAction.toDistribSMul.{u3, u2} A B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u3, u3, u2, max u2 u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6 (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) (AlgHom.algHomClass.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6))))) (Polynomial.aeval.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6 x) p) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommMonoidWithZero.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommSemiring.toCommMonoidWithZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommRing.toCommSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) _inst_3)))))) -> (IsIntegral.{u2, u2} (Subtype.{succ u2} B (fun (x : B) => Membership.mem.{u2, u2} B (Subalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) B (Subalgebra.instSetLikeSubalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7)) x (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p)))))) B (Subalgebra.toCommRing.{u1, u2} R B _inst_1 _inst_3 _inst_7 (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p))))) (CommRing.toRing.{u2} B _inst_3) (Subalgebra.toAlgebra.{u2, u1, u2} B R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u2} B _inst_3) (Ring.toSemiring.{u2} B (CommRing.toRing.{u2} B _inst_3)) _inst_7 (Algebra.id.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p))))) x)
+  forall {R : Type.{u1}} {A : Type.{u3}} {B : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u2} B] [_inst_6 : Algebra.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_7 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] (x : B) {p : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))}, (Polynomial.Monic.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) p) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (fun (_x : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) _x) (SMulHomClass.toFunLike.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (SMulZeroClass.toSMul.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddMonoid.toZero.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (DistribSMul.toSMulZeroClass.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddMonoid.toAddZeroClass.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (DistribMulAction.toDistribSMul.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (SMulZeroClass.toSMul.{u3, u2} A B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (DistribSMul.toSMulZeroClass.{u3, u2} A B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (DistribMulAction.toDistribSMul.{u3, u2} A B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u3, u3, u2, max u2 u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6 (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) (AlgHom.algHomClass.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6))))) (Polynomial.aeval.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6 x) p) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommMonoidWithZero.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommSemiring.toCommMonoidWithZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommRing.toCommSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2187 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) _inst_3)))))) -> (IsIntegral.{u2, u2} (Subtype.{succ u2} B (fun (x : B) => Membership.mem.{u2, u2} B (Subalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) B (Subalgebra.instSetLikeSubalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7)) x (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p)))))) B (Subalgebra.toCommRing.{u1, u2} R B _inst_1 _inst_3 _inst_7 (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p))))) (CommRing.toRing.{u2} B _inst_3) (Subalgebra.toAlgebra.{u2, u1, u2} B R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u2} B _inst_3) (Ring.toSemiring.{u2} B (CommRing.toRing.{u2} B _inst_3)) _inst_7 (Algebra.id.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p))))) x)
 Case conversion may be inaccurate. Consider using '#align is_integral_trans_aux isIntegral_trans_auxₓ'. -/
 theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x p = 0) :
     IsIntegral (adjoin R (↑(p.map <| algebraMap A B).frange : Set B)) x :=

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

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 
 ! This file was ported from Lean 3 source module ring_theory.integral_closure
-! leanprover-community/mathlib commit 641b6a82006416ec431b2987b354af9311fed4f2
+! leanprover-community/mathlib commit 38df578a6450a8c5142b3727e3ae894c2300cae0
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -20,6 +20,9 @@ import Mathbin.RingTheory.TensorProduct
 /-!
 # Integral closure of a subring.
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 If A is an R-algebra then `a : A` is integral over R if it is a root of a monic polynomial
 with coefficients in R. Enough theory is developed to prove that integral elements
 form a sub-R-algebra of A.

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

@@ -48,44 +48,70 @@ variable {R S A : Type _}
 
 variable [CommRing R] [Ring A] [Ring S] (f : R →+* S)
 
+#print RingHom.IsIntegralElem /-
 /-- An element `x` of `A` is said to be integral over `R` with respect to `f`
 if it is a root of a monic polynomial `p : R[X]` evaluated under `f` -/
 def RingHom.IsIntegralElem (f : R →+* A) (x : A) :=
   ∃ p : R[X], Monic p ∧ eval₂ f x p = 0
 #align ring_hom.is_integral_elem RingHom.IsIntegralElem
+-/
 
+#print RingHom.IsIntegral /-
 /-- A ring homomorphism `f : R →+* A` is said to be integral
 if every element `A` is integral with respect to the map `f` -/
 def RingHom.IsIntegral (f : R →+* A) :=
   ∀ x : A, f.IsIntegralElem x
 #align ring_hom.is_integral RingHom.IsIntegral
+-/
 
 variable [Algebra R A] (R)
 
+#print IsIntegral /-
 /-- An element `x` of an algebra `A` over a commutative ring `R` is said to be *integral*,
 if it is a root of some monic polynomial `p : R[X]`.
 Equivalently, the element is integral over `R` with respect to the induced `algebra_map` -/
 def IsIntegral (x : A) : Prop :=
   (algebraMap R A).IsIntegralElem x
 #align is_integral IsIntegral
+-/
 
 variable (A)
 
+#print Algebra.IsIntegral /-
 /-- An algebra is integral if every element of the extension is integral over the base ring -/
 protected def Algebra.IsIntegral : Prop :=
   (algebraMap R A).IsIntegral
 #align algebra.is_integral Algebra.IsIntegral
+-/
 
 variable {R A}
 
+/- warning: ring_hom.is_integral_map -> RingHom.is_integral_map is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_map RingHom.is_integral_mapₓ'. -/
 theorem RingHom.is_integral_map {x : R} : f.IsIntegralElem (f x) :=
   ⟨X - C x, monic_X_sub_C _, by simp⟩
 #align ring_hom.is_integral_map RingHom.is_integral_map
 
+/- warning: is_integral_algebra_map -> isIntegral_algebraMap is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align is_integral_algebra_map isIntegral_algebraMapₓ'. -/
 theorem isIntegral_algebraMap {x : R} : IsIntegral R (algebraMap R A x) :=
   (algebraMap R A).is_integral_map
 #align is_integral_algebra_map isIntegral_algebraMap
 
+/- warning: is_integral_of_noetherian -> isIntegral_of_noetherian is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align is_integral_of_noetherian isIntegral_of_noetherianₓ'. -/
 theorem isIntegral_of_noetherian (H : IsNoetherian R A) (x : A) : IsIntegral R x :=
   by
   let leval : R[X] →ₗ[R] A := (aeval x).toLinearMap
@@ -106,6 +132,12 @@ theorem isIntegral_of_noetherian (H : IsNoetherian R A) (x : A) : IsIntegral R x
   rw [LinearMap.map_sub, hpe, sub_self]
 #align is_integral_of_noetherian isIntegral_of_noetherian
 
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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A _inst_2)))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_4)))))))) (Subalgebra.toSubmodule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_4) S))) -> (forall (x : A), (Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_4) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_4) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_2) _inst_4)) x S) -> (IsIntegral.{u1, u2} R A _inst_1 _inst_2 _inst_4 x))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : Ring.{u1} A] [_inst_4 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2)] (S : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4), (IsNoetherian.{u2, u1} R (Subtype.{succ u1} A (fun (x : A) => Membership.mem.{u1, u1} A ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) S) (SetLike.instMembership.{u1, u1} ((fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) S) A (Submodule.setLike.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4))) x (FunLike.coe.{succ u1, succ u1, succ u1} (OrderEmbedding.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (SetLike.instPartialOrder.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)))) (Preorder.toLE.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4))))))) (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (fun (a : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) a) (RelHomClass.toFunLike.{u1, u1, u1} (OrderEmbedding.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (SetLike.instPartialOrder.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) A 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(OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4))))))) (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) => LE.le.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (SetLike.instPartialOrder.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) => LE.le.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A 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u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) => LE.le.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (SetLike.instPartialOrder.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) => LE.le.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) => LE.le.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (SetLike.instPartialOrder.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)))) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) => LE.le.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (Ring.toSemiring.{u1} A _inst_2)))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) S))) -> (forall (x : A), (Membership.mem.{u1, u1} A (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) (SetLike.instMembership.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u1} A _inst_2) _inst_4)) x S) -> (IsIntegral.{u2, u1} R A _inst_1 _inst_2 _inst_4 x))
+Case conversion may be inaccurate. Consider using '#align is_integral_of_submodule_noetherian isIntegral_of_submodule_noetherianₓ'. -/
 theorem isIntegral_of_submodule_noetherian (S : Subalgebra R A) (H : IsNoetherian R S.toSubmodule)
     (x : A) (hx : x ∈ S) : IsIntegral R x :=
   by
@@ -131,6 +163,12 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S]
 
 variable [Algebra R A] [Algebra R B] (f : R →+* S)
 
+/- warning: map_is_integral -> map_isIntegral is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {B : Type.{u3}} {C : Type.{u4}} {F : Type.{u5}} [_inst_7 : Ring.{u3} B] [_inst_8 : Ring.{u4} C] [_inst_9 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B _inst_7)] [_inst_10 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7)] [_inst_11 : Algebra.{u1, u4} R C (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} C _inst_8)] [_inst_12 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B _inst_7) _inst_9)))))] [_inst_13 : Algebra.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8)] [_inst_14 : IsScalarTower.{u1, u2, u4} R A C (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u4} A C (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (SMulWithZero.toSmulZeroClass.{u2, u4} A C (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (MulActionWithZero.toSMulWithZero.{u2, u4} A C (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (Module.toMulActionWithZero.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13))))) (SMulZeroClass.toHasSmul.{u1, u4} R C (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (SMulWithZero.toSmulZeroClass.{u1, u4} R C (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.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (MulActionWithZero.toSMulWithZero.{u1, u4} R C (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (Module.toMulActionWithZero.{u1, u4} R C (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} R C (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} C _inst_8) _inst_11)))))] {b : B} [_inst_15 : AlgHomClass.{u5, u2, u3, u4} F A B C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13] (f : F), (IsIntegral.{u1, u3} R B _inst_1 _inst_7 _inst_9 b) -> (IsIntegral.{u1, u4} R C _inst_1 _inst_8 _inst_11 (coeFn.{succ u5, max (succ u3) (succ u4)} F (fun (_x : F) => B -> C) (FunLike.hasCoeToFun.{succ u5, succ u3, succ u4} F B (fun (_x : B) => C) (SMulHomClass.toFunLike.{u5, u2, u3, u4} F A B C (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))))) (DistribSMul.toSmulZeroClass.{u2, u3} A B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))))) (DistribMulAction.toDistribSMul.{u2, u3} A B (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))) (Module.toDistribMulAction.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10))))) (SMulZeroClass.toHasSmul.{u2, u4} A C (AddZeroClass.toHasZero.{u4} C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))))) (DistribSMul.toSmulZeroClass.{u2, u4} A C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribMulAction.toDistribSMul.{u2, u4} A C (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13))))) (DistribMulActionHomClass.toSmulHomClass.{u5, u2, u3, u4} F A B C (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u5, u2, u3, u4} F A B C (MonoidWithZero.toMonoid.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))) (Module.toDistribMulAction.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B _inst_7)))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u2, u4} A C (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u2, u4} A C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (AlgHom.nonUnitalAlgHomClass.{u2, u3, u4, u5} A B C (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13 F _inst_15))))) f b))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {B : Type.{u5}} {C : Type.{u4}} {F : Type.{u3}} [_inst_7 : Ring.{u5} B] [_inst_8 : Ring.{u4} C] [_inst_9 : Algebra.{u2, u5} R B (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u5} B _inst_7)] [_inst_10 : Algebra.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7)] [_inst_11 : Algebra.{u2, u4} R C (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u4} C _inst_8)] [_inst_12 : IsScalarTower.{u2, u1, u5} R A B (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.toSMul.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10) (Algebra.toSMul.{u2, u5} R B (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u5} B _inst_7) _inst_9)] [_inst_13 : Algebra.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8)] [_inst_14 : IsScalarTower.{u2, u1, u4} R A C (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.toSMul.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13) (Algebra.toSMul.{u2, u4} R C (CommRing.toCommSemiring.{u2} R _inst_1) (Ring.toSemiring.{u4} C _inst_8) _inst_11)] {b : B} [_inst_15 : AlgHomClass.{u3, u1, u5, u4} F A B C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13] (f : F), (IsIntegral.{u2, u5} R B _inst_1 _inst_7 _inst_9 b) -> (IsIntegral.{u2, u4} R ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : B) => C) b) _inst_1 _inst_8 _inst_11 (FunLike.coe.{succ u3, succ u5, succ u4} F B (fun (_x : B) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : B) => C) _x) (SMulHomClass.toFunLike.{u3, u1, u5, u4} F A B C (SMulZeroClass.toSMul.{u1, u5} A B (AddMonoid.toZero.{u5} B (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u5} A B (AddMonoid.toAddZeroClass.{u5} B (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u5} A B (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10))))) (SMulZeroClass.toSMul.{u1, u4} A C (AddMonoid.toZero.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u4} A C (AddMonoid.toAddZeroClass.{u4} C (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u4} A C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13))))) (DistribMulActionHomClass.toSMulHomClass.{u3, u1, u5, u4} F A B C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u5} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))))) (AddCommMonoid.toAddMonoid.{u4} C (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{u3, u1, u5, u4} F A B C (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8))) (Module.toDistribMulAction.{u1, u5} A B (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} B (Semiring.toNonAssocSemiring.{u5} B (Ring.toSemiring.{u5} B _inst_7)))) (Algebra.toModule.{u1, u5} A B (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) _inst_10)) (Module.toDistribMulAction.{u1, u4} A C (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} C (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} C (Semiring.toNonAssocSemiring.{u4} C (Ring.toSemiring.{u4} C _inst_8)))) (Algebra.toModule.{u1, u4} A C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u4} C _inst_8) _inst_13)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u5, u4, u3} A B C (CommRing.toCommSemiring.{u1} A _inst_2) (Ring.toSemiring.{u5} B _inst_7) (Ring.toSemiring.{u4} C _inst_8) _inst_10 _inst_13 F _inst_15)))) f b))
+Case conversion may be inaccurate. Consider using '#align map_is_integral map_isIntegralₓ'. -/
 theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
     [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B} [AlgHomClass F A B C] (f : F)
     (hb : IsIntegral R b) : IsIntegral R (f b) :=
@@ -141,6 +179,12 @@ theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra
     aeval_alg_hom_apply, aeval_map_algebra_map, aeval_def, hP.2, _root_.map_zero]
 #align map_is_integral map_isIntegral
 
+/- warning: is_integral_map_of_comp_eq_of_is_integral -> isIntegral_map_of_comp_eq_of_isIntegral is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} {T : Type.{u3}} {U : Type.{u4}} [_inst_7 : CommRing.{u1} R] [_inst_8 : CommRing.{u2} S] [_inst_9 : CommRing.{u3} T] [_inst_10 : CommRing.{u4} U] [_inst_11 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_7) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_8))] [_inst_12 : Algebra.{u3, u4} T U (CommRing.toCommSemiring.{u3} T _inst_9) (Ring.toSemiring.{u4} U (CommRing.toRing.{u4} U _inst_10))] (φ : RingHom.{u1, u3} R T (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_7))) (NonAssocRing.toNonAssocSemiring.{u3} T (Ring.toNonAssocRing.{u3} T (CommRing.toRing.{u3} T _inst_9)))) (ψ : RingHom.{u2, u4} S U (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_8))) (NonAssocRing.toNonAssocSemiring.{u4} U (Ring.toNonAssocRing.{u4} U (CommRing.toRing.{u4} U _inst_10)))), (Eq.{max (succ u1) (succ u4)} (RingHom.{u1, u4} R U (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_7))) (Semiring.toNonAssocSemiring.{u4} U (Ring.toSemiring.{u4} U (CommRing.toRing.{u4} U _inst_10)))) (RingHom.comp.{u1, u3, u4} R T U (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_7))) (Semiring.toNonAssocSemiring.{u3} T (CommSemiring.toSemiring.{u3} T (CommRing.toCommSemiring.{u3} T _inst_9))) (Semiring.toNonAssocSemiring.{u4} U (Ring.toSemiring.{u4} U (CommRing.toRing.{u4} U _inst_10))) (algebraMap.{u3, u4} T U (CommRing.toCommSemiring.{u3} T _inst_9) (Ring.toSemiring.{u4} U (CommRing.toRing.{u4} U _inst_10)) _inst_12) φ) (RingHom.comp.{u1, u2, u4} R S U (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_7))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_8))) (NonAssocRing.toNonAssocSemiring.{u4} U (Ring.toNonAssocRing.{u4} U (CommRing.toRing.{u4} U _inst_10))) ψ (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_7) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_8)) _inst_11))) -> (forall {a : S}, (IsIntegral.{u1, u2} R S _inst_7 (CommRing.toRing.{u2} S _inst_8) _inst_11 a) -> (IsIntegral.{u3, u4} T U _inst_9 (CommRing.toRing.{u4} U _inst_10) _inst_12 (coeFn.{max (succ u2) (succ u4), max (succ u2) (succ u4)} (RingHom.{u2, u4} S U (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_8))) (NonAssocRing.toNonAssocSemiring.{u4} U (Ring.toNonAssocRing.{u4} U (CommRing.toRing.{u4} U _inst_10)))) (fun (_x : RingHom.{u2, u4} S U (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_8))) (NonAssocRing.toNonAssocSemiring.{u4} U (Ring.toNonAssocRing.{u4} U (CommRing.toRing.{u4} U _inst_10)))) => S -> U) (RingHom.hasCoeToFun.{u2, u4} S U (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_8))) (NonAssocRing.toNonAssocSemiring.{u4} U (Ring.toNonAssocRing.{u4} U (CommRing.toRing.{u4} U _inst_10)))) ψ a)))
+but is expected to have type
+  forall {R : Type.{u4}} {S : Type.{u3}} {T : Type.{u2}} {U : Type.{u1}} [_inst_7 : CommRing.{u4} R] [_inst_8 : CommRing.{u3} S] [_inst_9 : CommRing.{u2} T] [_inst_10 : CommRing.{u1} U] [_inst_11 : Algebra.{u4, u3} R S (CommRing.toCommSemiring.{u4} R _inst_7) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8))] [_inst_12 : Algebra.{u2, u1} T U (CommRing.toCommSemiring.{u2} T _inst_9) (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10))] (φ : RingHom.{u4, u2} R T (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R (CommRing.toCommSemiring.{u4} R _inst_7))) (Semiring.toNonAssocSemiring.{u2} T (CommSemiring.toSemiring.{u2} T (CommRing.toCommSemiring.{u2} T _inst_9)))) (ψ : RingHom.{u3, u1} S U (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8))) (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10)))), (Eq.{max (succ u4) (succ u1)} (RingHom.{u4, u1} R U (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R (CommRing.toCommSemiring.{u4} R _inst_7))) (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10)))) (RingHom.comp.{u4, u2, u1} R T U (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R (CommRing.toCommSemiring.{u4} R _inst_7))) (Semiring.toNonAssocSemiring.{u2} T (CommSemiring.toSemiring.{u2} T (CommRing.toCommSemiring.{u2} T _inst_9))) (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10))) (algebraMap.{u2, u1} T U (CommRing.toCommSemiring.{u2} T _inst_9) (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10)) _inst_12) φ) (RingHom.comp.{u4, u3, u1} R S U (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R (CommRing.toCommSemiring.{u4} R _inst_7))) (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8))) (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10))) ψ (algebraMap.{u4, u3} R S (CommRing.toCommSemiring.{u4} R _inst_7) (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8)) _inst_11))) -> (forall {a : S}, (IsIntegral.{u4, u3} R S _inst_7 (CommRing.toRing.{u3} S _inst_8) _inst_11 a) -> (IsIntegral.{u2, u1} T ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : S) => U) a) _inst_9 (CommRing.toRing.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : S) => U) a) _inst_10) _inst_12 (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (RingHom.{u3, u1} S U (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8))) (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10)))) S (fun (_x : S) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : S) => U) _x) (MulHomClass.toFunLike.{max u3 u1, u3, u1} (RingHom.{u3, u1} S U (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8))) (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10)))) S U (NonUnitalNonAssocSemiring.toMul.{u3} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8))))) (NonUnitalNonAssocSemiring.toMul.{u1} U (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} U (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u1, u3, u1} (RingHom.{u3, u1} S U (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8))) (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10)))) S U (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} U (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u1, u3, u1} (RingHom.{u3, u1} S U (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8))) (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10)))) S U (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8))) (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10))) (RingHom.instRingHomClassRingHom.{u3, u1} S U (Semiring.toNonAssocSemiring.{u3} S (CommSemiring.toSemiring.{u3} S (CommRing.toCommSemiring.{u3} S _inst_8))) (Semiring.toNonAssocSemiring.{u1} U (CommSemiring.toSemiring.{u1} U (CommRing.toCommSemiring.{u1} U _inst_10))))))) ψ a)))
+Case conversion may be inaccurate. Consider using '#align is_integral_map_of_comp_eq_of_is_integral isIntegral_map_of_comp_eq_of_isIntegralₓ'. -/
 theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R] [CommRing S]
     [CommRing T] [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
     (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
@@ -152,6 +196,12 @@ theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R]
     RingHom.map_zero]
 #align is_integral_map_of_comp_eq_of_is_integral isIntegral_map_of_comp_eq_of_isIntegral
 
+/- warning: is_integral_alg_hom_iff -> isIntegral_algHom_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {A : Type.{u2}} {B : Type.{u3}} [_inst_7 : Ring.{u2} A] [_inst_8 : Ring.{u3} B] [_inst_9 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7)] [_inst_10 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B _inst_8)] (f : AlgHom.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10), (Function.Injective.{succ u2, succ u3} A B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (AlgHom.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) (fun (_x : AlgHom.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) => A -> B) ([anonymous].{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) f)) -> (forall {x : A}, Iff (IsIntegral.{u1, u3} R B _inst_1 _inst_8 _inst_10 (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (AlgHom.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) (fun (_x : AlgHom.{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) => A -> B) ([anonymous].{u1, u2, u3} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A _inst_7) (Ring.toSemiring.{u3} B _inst_8) _inst_9 _inst_10) f x)) (IsIntegral.{u1, u2} R A _inst_1 _inst_7 _inst_9 x))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {A : Type.{u3}} {B : Type.{u2}} [_inst_7 : Ring.{u3} A] [_inst_8 : Ring.{u2} B] [_inst_9 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7)] [_inst_10 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8)] (f : AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10), (Function.Injective.{succ u3, succ u2} A B (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9))))) (SMulZeroClass.toSMul.{u1, u2} R B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u2} R B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u2} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B 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(Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) (AlgHom.algHomClass.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10))))) f)) -> (forall {x : A}, Iff (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) x) _inst_1 _inst_8 _inst_10 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => B) _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9))))) (SMulZeroClass.toSMul.{u1, u2} R B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribSMul.toSMulZeroClass.{u1, u2} R B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))))) (DistribMulAction.toDistribSMul.{u1, u2} R B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (Ring.toSemiring.{u3} A _inst_7)))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) _inst_9)) (Module.toDistribMulAction.{u1, u2} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (Ring.toSemiring.{u2} B _inst_8)))) (Algebra.toModule.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} B _inst_8) _inst_10)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgHom.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) (AlgHom.algHomClass.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10))))) f x)) (IsIntegral.{u1, u3} R A _inst_1 _inst_7 _inst_9 x))
+Case conversion may be inaccurate. Consider using '#align is_integral_alg_hom_iff isIntegral_algHom_iffₓ'. -/
 theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A →ₐ[R] B) (hf : Function.Injective f) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
   by
@@ -162,27 +212,57 @@ theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Al
     map_eq_zero_iff f hf] at hx
 #align is_integral_alg_hom_iff isIntegral_algHom_iff
 
+/- warning: is_integral_alg_equiv -> isIntegral_algEquiv is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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(AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) (AlgEquivClass.toAlgHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10) R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10 (AlgEquiv.instAlgEquivClassAlgEquiv.{u1, u3, u2} R A B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} A _inst_7) (Ring.toSemiring.{u2} B _inst_8) _inst_9 _inst_10)))))) f x)) (IsIntegral.{u1, u3} R A _inst_1 _inst_7 _inst_9 x)
+Case conversion may be inaccurate. Consider using '#align is_integral_alg_equiv isIntegral_algEquivₓ'. -/
 @[simp]
 theorem isIntegral_algEquiv {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
   ⟨fun h => by simpa using map_isIntegral f.symm.to_alg_hom h, map_isIntegral f.toAlgHom⟩
 #align is_integral_alg_equiv isIntegral_algEquiv
 
+/- warning: is_integral_of_is_scalar_tower -> isIntegral_of_isScalarTower is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_6 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_7 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_8 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6)))))] {x : B}, (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_6 x) -> (IsIntegral.{u2, u3} A B _inst_2 (CommRing.toRing.{u3} B _inst_3) _inst_7 x)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u3}} {B : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u2} B] [_inst_5 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_6 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_7 : Algebra.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_8 : IsScalarTower.{u1, u3, u2} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_5) (Algebra.toSMul.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) (Algebra.toSMul.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)] {x : B}, (IsIntegral.{u1, u2} R B _inst_1 (CommRing.toRing.{u2} B _inst_3) _inst_6 x) -> (IsIntegral.{u3, u2} A B _inst_2 (CommRing.toRing.{u2} B _inst_3) _inst_7 x)
+Case conversion may be inaccurate. Consider using '#align is_integral_of_is_scalar_tower isIntegral_of_isScalarTowerₓ'. -/
 theorem isIntegral_of_isScalarTower [Algebra A B] [IsScalarTower R A B] {x : B}
     (hx : IsIntegral R x) : IsIntegral A x :=
   let ⟨p, hp, hpx⟩ := hx
   ⟨p.map <| algebraMap R A, hp.map _, by rw [← aeval_def, aeval_map_algebra_map, aeval_def, hpx]⟩
 #align is_integral_of_is_scalar_tower isIntegral_of_isScalarTower
 
+/- warning: map_is_integral_int -> map_isIntegral_int is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align map_is_integral_int map_isIntegral_intₓ'. -/
 theorem map_isIntegral_int {B C F : Type _} [Ring B] [Ring C] {b : B} [RingHomClass F B C] (f : F)
     (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
   map_isIntegral (f : B →+* C).toIntAlgHom hb
 #align map_is_integral_int map_isIntegral_int
 
+/- warning: is_integral_of_subring -> isIntegral_ofSubring is a dubious translation:
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+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {x : A} (T : Subring.{u2} R (CommRing.toRing.{u2} R _inst_1)), (IsIntegral.{u2, u1} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Subring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (SetLike.instMembership.{u2, u2} (Subring.{u2} R (CommRing.toRing.{u2} R _inst_1)) R (Subring.instSetLikeSubring.{u2} R (CommRing.toRing.{u2} R _inst_1))) x T)) A (Subring.toCommRing.{u2} R _inst_1 T) (CommRing.toRing.{u1} A _inst_2) (Algebra.ofSubring.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 T) x) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x)
+Case conversion may be inaccurate. Consider using '#align is_integral_of_subring isIntegral_ofSubringₓ'. -/
 theorem isIntegral_ofSubring {x : A} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
   isIntegral_of_isScalarTower hx
 #align is_integral_of_subring isIntegral_ofSubring
 
+/- warning: is_integral.algebra_map -> IsIntegral.algebraMap is a dubious translation:
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+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_6 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_7 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_8 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (MulZeroClass.toHasZero.{u1} R 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B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7) x))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u3}} {B : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u2} B] [_inst_5 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_6 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_7 : Algebra.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_8 : IsScalarTower.{u1, u3, u2} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_5) (Algebra.toSMul.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) (Algebra.toSMul.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)] {x : A}, (IsIntegral.{u1, u3} R A _inst_1 (CommRing.toRing.{u3} A _inst_2) _inst_5 x) -> (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_1 (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_3) _inst_6 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonUnitalNonAssocSemiring.toMul.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) x))
+Case conversion may be inaccurate. Consider using '#align is_integral.algebra_map IsIntegral.algebraMapₓ'. -/
 theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A} (h : IsIntegral R x) :
     IsIntegral R (algebraMap A B x) :=
   by
@@ -191,12 +271,24 @@ theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A} (h : I
   rw [IsScalarTower.algebraMap_eq R A B, ← hom_eval₂, hx, RingHom.map_zero]
 #align is_integral.algebra_map IsIntegral.algebraMap
 
+/- warning: is_integral_algebra_map_iff -> isIntegral_algebraMap_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_6 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_7 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_8 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6)))))] {x : A}, (Function.Injective.{succ u2, succ u3} A B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7))) -> (Iff (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_6 (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7) x)) (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u3}} {B : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u2} B] [_inst_5 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_6 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_7 : Algebra.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_8 : IsScalarTower.{u1, u3, u2} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_5) (Algebra.toSMul.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) (Algebra.toSMul.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)] {x : A}, (Function.Injective.{succ u3, succ u2} A B (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonUnitalNonAssocSemiring.toMul.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7))) -> (Iff (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_1 (CommRing.toRing.{u2} ((fun 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) x)) (IsIntegral.{u1, u3} R A _inst_1 (CommRing.toRing.{u3} A _inst_2) _inst_5 x))
+Case conversion may be inaccurate. Consider using '#align is_integral_algebra_map_iff isIntegral_algebraMap_iffₓ'. -/
 theorem isIntegral_algebraMap_iff [Algebra A B] [IsScalarTower R A B] {x : A}
     (hAB : Function.Injective (algebraMap A B)) :
     IsIntegral R (algebraMap A B x) ↔ IsIntegral R x :=
   isIntegral_algHom_iff (IsScalarTower.toAlgHom R A B) hAB
 #align is_integral_algebra_map_iff isIntegral_algebraMap_iff
 
+/- warning: is_integral_iff_is_integral_closure_finite -> isIntegral_iff_isIntegral_closure_finite is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {r : A}, Iff (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 r) (Exists.{succ u1} (Set.{u1} R) (fun (s : Set.{u1} R) => And (Set.Finite.{u1} R s) (IsIntegral.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Subring.{u1} R (CommRing.toRing.{u1} R _inst_1)) Type.{u1} (SetLike.hasCoeToSort.{u1, u1} (Subring.{u1} R (CommRing.toRing.{u1} R _inst_1)) R (Subring.setLike.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s)) A (Subring.toCommRing.{u1} R _inst_1 (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s)) (CommRing.toRing.{u2} A _inst_2) (Algebra.ofSubring.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (Subring.closure.{u1} R (CommRing.toRing.{u1} R _inst_1) s)) r)))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {r : A}, Iff (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 r) (Exists.{succ u2} (Set.{u2} R) (fun (s : Set.{u2} R) => And (Set.Finite.{u2} R s) (IsIntegral.{u2, u1} (Subtype.{succ u2} R (fun (x : R) => Membership.mem.{u2, u2} R (Subring.{u2} R (CommRing.toRing.{u2} R _inst_1)) (SetLike.instMembership.{u2, u2} (Subring.{u2} R (CommRing.toRing.{u2} R _inst_1)) R (Subring.instSetLikeSubring.{u2} R (CommRing.toRing.{u2} R _inst_1))) x (Subring.closure.{u2} R (CommRing.toRing.{u2} R _inst_1) s))) A (Subring.toCommRing.{u2} R _inst_1 (Subring.closure.{u2} R (CommRing.toRing.{u2} R _inst_1) s)) (CommRing.toRing.{u1} A _inst_2) (Algebra.ofSubring.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (Subring.closure.{u2} R (CommRing.toRing.{u2} R _inst_1) s)) r)))
+Case conversion may be inaccurate. Consider using '#align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finiteₓ'. -/
 theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
     IsIntegral R r ↔ ∃ s : Set R, s.Finite ∧ IsIntegral (Subring.closure s) r :=
   by
@@ -208,7 +300,13 @@ theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
   exact isIntegral_ofSubring _ hsr
 #align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finite
 
-theorem fG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
+/- warning: fg_adjoin_singleton_of_integral -> FG_adjoin_singleton_of_integral is a dubious translation:
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(Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (LE.le.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)))))))) (Subalgebra.toSubmodule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Algebra.adjoin.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) x))))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (x : A), (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x) -> (Submodule.FG.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (FunLike.coe.{succ u1, succ u1, succ u1} (OrderEmbedding.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} 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(CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)))) (Preorder.toLE.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} 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(Subalgebra.toSubmodule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.adjoin.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5 (Singleton.singleton.{u1, u1} A (Set.{u1} A) (Set.instSingletonSet.{u1} A) x))))
+Case conversion may be inaccurate. Consider using '#align fg_adjoin_singleton_of_integral FG_adjoin_singleton_of_integralₓ'. -/
+theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
     (Algebra.adjoin R ({x} : Set A)).toSubmodule.FG :=
   by
   rcases hx with ⟨f, hfm, hfx⟩
@@ -237,9 +335,15 @@ theorem fG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rw [degree_le_iff_coeff_zero] at this
   rw [mem_support_iff] at hkq; apply hkq; apply this
   exact lt_of_le_of_lt degree_le_nat_degree (WithBot.coe_lt_coe.2 hk)
-#align fg_adjoin_singleton_of_integral fG_adjoin_singleton_of_integral
-
-theorem fG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
+#align fg_adjoin_singleton_of_integral FG_adjoin_singleton_of_integral
+
+/- warning: fg_adjoin_of_finite -> FG_adjoin_of_finite is a dubious translation:
+lean 3 declaration is
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_inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))))) (fun (_x : RelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (LE.le.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)))))))) => (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) -> (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))) (RelEmbedding.hasCoeToFun.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (LE.le.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (Preorder.toHasLe.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)))))))) (Subalgebra.toSubmodule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Algebra.adjoin.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5 s)))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] {s : Set.{u2} A}, (Set.Finite.{u2} A s) -> (forall (x : A), (Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x s) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x)) -> (Submodule.FG.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) 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(NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R 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(fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Algebra.adjoin.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5 s)))
+Case conversion may be inaccurate. Consider using '#align fg_adjoin_of_finite FG_adjoin_of_finiteₓ'. -/
+theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
     (Algebra.adjoin R s).toSubmodule.FG :=
   Set.Finite.induction_on hfs
     (fun _ =>
@@ -253,19 +357,31 @@ theorem fG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
       rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule] <;>
         exact
           fg.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)
-            (fG_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s))
+            (FG_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s))
     his
-#align fg_adjoin_of_finite fG_adjoin_of_finite
-
+#align fg_adjoin_of_finite FG_adjoin_of_finite
+
+/- warning: is_noetherian_adjoin_finset -> isNoetherian_adjoin_finset is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_7 : IsNoetherianRing.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))] (s : Finset.{u2} A), (forall (x : A), (Membership.Mem.{u2, u2} A (Finset.{u2} A) (Finset.hasMem.{u2} A) x s) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x)) -> (IsNoetherian.{u1, u2} R (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A 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(Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5 ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} A) (Set.{u2} A) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} A) (Set.{u2} A) (CoeTCₓ.coe.{succ u2, succ u2} (Finset.{u2} A) (Set.{u2} A) (Finset.Set.hasCoeT.{u2} A))) s))))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] [_inst_7 : IsNoetherianRing.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))] (s : Finset.{u1} A), (forall (x : A), (Membership.mem.{u1, u1} A (Finset.{u1} A) (Finset.instMembershipFinset.{u1} A) x s) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x)) -> (IsNoetherian.{u2, u1} R (Subtype.{succ u1} A (fun (x : A) => Membership.mem.{u1, u1} A (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (SetLike.instMembership.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) A 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(Subalgebra.instModuleSubtypeMemSubalgebraInstMembershipInstSetLikeSubalgebraToSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonAssocSemiringToNonAssocSemiringToSubsemiring.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5 (Algebra.adjoin.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5 (Finset.toSet.{u1} A s))))
+Case conversion may be inaccurate. Consider using '#align is_noetherian_adjoin_finset isNoetherian_adjoin_finsetₓ'. -/
 theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
     (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (↑s : Set A)) :=
-  isNoetherian_of_fg_of_noetherian _ (fG_adjoin_of_finite s.finite_toSet hs)
+  isNoetherian_of_fg_of_noetherian _ (FG_adjoin_of_finite s.finite_toSet hs)
 #align is_noetherian_adjoin_finset isNoetherian_adjoin_finset
 
+/- warning: is_integral_of_mem_of_fg -> isIntegral_of_mem_of_FG is a dubious translation:
+lean 3 declaration is
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(CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) x S) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] (S : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5), (Submodule.FG.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (FunLike.coe.{succ u1, succ u1, succ u1} (OrderEmbedding.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R 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R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)))) (Preorder.toLE.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5))))))) (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (fun (_x : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) => Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) _x) (RelHomClass.toFunLike.{u1, u1, u1} (OrderEmbedding.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (SetLike.instPartialOrder.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A 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(CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5))))))) (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) => LE.le.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (SetLike.instPartialOrder.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)))) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) => LE.le.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) => LE.le.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (SetLike.instPartialOrder.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)))) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) => LE.le.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (Preorder.toLE.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) S)) -> (forall (x : A), (Membership.mem.{u1, u1} A (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (SetLike.instMembership.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) x S) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x))
+Case conversion may be inaccurate. Consider using '#align is_integral_of_mem_of_fg isIntegral_of_mem_of_FGₓ'. -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
   then all elements of `S` are integral over `R`. -/
-theorem isIntegral_of_mem_of_fG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) :
+theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) :
     IsIntegral R x :=
   by
   -- say `x ∈ S`. We want to prove that `x` is integral over `R`.
@@ -362,13 +478,25 @@ theorem isIntegral_of_mem_of_fG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   change (⟨_, this⟩ : S₀) • r ∈ _
   rw [Finsupp.mem_supported] at hlx1
   exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _
-#align is_integral_of_mem_of_fg isIntegral_of_mem_of_fG
-
+#align is_integral_of_mem_of_fg isIntegral_of_mem_of_FG
+
+/- warning: module.End.is_integral -> Module.End.isIntegral is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : Type.{u2}} [_inst_7 : AddCommGroup.{u2} M] [_inst_8 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_7)] [_inst_9 : Module.Finite.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_7) _inst_8], Algebra.IsIntegral.{u1, u2} R (Module.End.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_7) _inst_8) _inst_1 (Module.End.ring.{u1, u2} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) _inst_7 _inst_8) (Module.End.algebra.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_7) _inst_8)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {M : Type.{u2}} [_inst_7 : AddCommGroup.{u2} M] [_inst_8 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_7)] [_inst_9 : Module.Finite.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_7) _inst_8], Algebra.IsIntegral.{u1, u2} R (Module.End.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} M _inst_7) _inst_8) _inst_1 (Module.End.ring.{u1, u2} R M (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) _inst_7 _inst_8) (Module.instAlgebraEndToSemiringSemiring.{u1, u2} R M (CommRing.toCommSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_7) _inst_8)
+Case conversion may be inaccurate. Consider using '#align module.End.is_integral Module.End.isIntegralₓ'. -/
 theorem Module.End.isIntegral {M : Type _} [AddCommGroup M] [Module R M] [Module.Finite R M] :
     Algebra.IsIntegral R (Module.End R M) :=
   LinearMap.exists_monic_and_aeval_eq_zero R
 #align module.End.is_integral Module.End.isIntegral
 
+/- warning: is_integral_of_smul_mem_submodule -> isIntegral_of_smul_mem_submodule is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {M : Type.{u3}} [_inst_7 : AddCommGroup.{u3} M] [_inst_8 : Module.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)] [_inst_9 : Module.{u2, u3} A M (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)] [_inst_10 : IsScalarTower.{u1, u2, u3} R A M (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))))) (SMulZeroClass.toHasSmul.{u2, u3} A M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} A M (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} A M (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (Module.toMulActionWithZero.{u2, u3} A M (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9)))) (SMulZeroClass.toHasSmul.{u1, u3} R M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (SMulWithZero.toSmulZeroClass.{u1, u3} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (MulActionWithZero.toSMulWithZero.{u1, u3} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (Module.toMulActionWithZero.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8))))] [_inst_11 : NoZeroSMulDivisors.{u2, u3} A M (MulZeroClass.toHasZero.{u2} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (SubNegMonoid.toAddMonoid.{u3} M (AddGroup.toSubNegMonoid.{u3} M (AddCommGroup.toAddGroup.{u3} M _inst_7))))) (SMulZeroClass.toHasSmul.{u2, u3} A M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} A M (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} A M (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (Module.toMulActionWithZero.{u2, u3} A M (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9))))] (N : Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8), (Ne.{succ u3} (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) N (Bot.bot.{u3} (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (Submodule.hasBot.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8))) -> (Submodule.FG.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8 N) -> (forall (x : A), (forall (n : M), (Membership.Mem.{u3, u3} M (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) M (Submodule.setLike.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8)) n N) -> (Membership.Mem.{u3, u3} M (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) M (Submodule.setLike.{u1, u3} R M (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8)) (SMul.smul.{u2, u3} A M (SMulZeroClass.toHasSmul.{u2, u3} A M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (SMulWithZero.toSmulZeroClass.{u2, u3} A M (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (MulActionWithZero.toSMulWithZero.{u2, u3} A M (Semiring.toMonoidWithZero.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)))) (Module.toMulActionWithZero.{u2, u3} A M (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9)))) x n) N)) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {M : Type.{u3}} [_inst_7 : AddCommGroup.{u3} M] [_inst_8 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)] [_inst_9 : Module.{u1, u3} A M (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7)] [_inst_10 : IsScalarTower.{u2, u1, u3} R A M (Algebra.toSMul.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (SMulZeroClass.toSMul.{u1, u3} A M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u3} A M (CommMonoidWithZero.toZero.{u1} A (CommSemiring.toCommMonoidWithZero.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u3} A M (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (Module.toMulActionWithZero.{u1, u3} A M (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9)))) (SMulZeroClass.toSMul.{u2, u3} R M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (SMulWithZero.toSMulZeroClass.{u2, u3} R M (CommMonoidWithZero.toZero.{u2} R (CommSemiring.toCommMonoidWithZero.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (MulActionWithZero.toSMulWithZero.{u2, u3} R M (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (Module.toMulActionWithZero.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8))))] [_inst_11 : NoZeroSMulDivisors.{u1, u3} A M (CommMonoidWithZero.toZero.{u1} A (CommSemiring.toCommMonoidWithZero.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (SMulZeroClass.toSMul.{u1, u3} A M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u3} A M (CommMonoidWithZero.toZero.{u1} A (CommSemiring.toCommMonoidWithZero.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u3} A M (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (Module.toMulActionWithZero.{u1, u3} A M (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9))))] (N : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8), (Ne.{succ u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) N (Bot.bot.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (Submodule.instBotSubmodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8))) -> (Submodule.FG.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8 N) -> (forall (x : A), (forall (n : M), (Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8)) n N) -> (Membership.mem.{u3, u3} M (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) (SetLike.instMembership.{u3, u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8) M (Submodule.setLike.{u2, u3} R M (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_8)) (HSMul.hSMul.{u1, u3, u3} A M M (instHSMul.{u1, u3} A M (SMulZeroClass.toSMul.{u1, u3} A M (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (SMulWithZero.toSMulZeroClass.{u1, u3} A M (CommMonoidWithZero.toZero.{u1} A (CommSemiring.toCommMonoidWithZero.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (MulActionWithZero.toSMulWithZero.{u1, u3} A M (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (NegZeroClass.toZero.{u3} M (SubNegZeroMonoid.toNegZeroClass.{u3} M (SubtractionMonoid.toSubNegZeroMonoid.{u3} M (SubtractionCommMonoid.toSubtractionMonoid.{u3} M (AddCommGroup.toDivisionAddCommMonoid.{u3} M _inst_7))))) (Module.toMulActionWithZero.{u1, u3} A M (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (AddCommGroup.toAddCommMonoid.{u3} M _inst_7) _inst_9))))) x n) N)) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x))
+Case conversion may be inaccurate. Consider using '#align is_integral_of_smul_mem_submodule isIntegral_of_smul_mem_submoduleₓ'. -/
 /-- Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
 and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. -/
 theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R M] [Module A M]
@@ -410,14 +538,32 @@ theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R
 
 variable {f}
 
+/- warning: ring_hom.finite.to_is_integral -> RingHom.Finite.to_isIntegral is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] {f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))}, (RingHom.Finite.{u1, u2} R S _inst_1 _inst_4 f) -> (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f)
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_4 : CommRing.{u1} S] {f : 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_4)))}, (RingHom.Finite.{u2, u1} R S _inst_1 _inst_4 f) -> (RingHom.IsIntegral.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) f)
+Case conversion may be inaccurate. Consider using '#align ring_hom.finite.to_is_integral RingHom.Finite.to_isIntegralₓ'. -/
 theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
   letI := f.to_algebra
-  fun x => isIntegral_of_mem_of_fG ⊤ h.1 _ trivial
+  fun x => isIntegral_of_mem_of_FG ⊤ h.1 _ trivial
 #align ring_hom.finite.to_is_integral RingHom.Finite.to_isIntegral
 
+/- warning: ring_hom.is_integral.of_finite -> RingHom.IsIntegral.of_finite is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] {f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))}, (RingHom.Finite.{u1, u2} R S _inst_1 _inst_4 f) -> (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f)
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_4 : CommRing.{u1} S] {f : 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_4)))}, (RingHom.Finite.{u2, u1} R S _inst_1 _inst_4 f) -> (RingHom.IsIntegral.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) f)
+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral.of_finite RingHom.IsIntegral.of_finiteₓ'. -/
 alias RingHom.Finite.to_isIntegral ← RingHom.IsIntegral.of_finite
 #align ring_hom.is_integral.of_finite RingHom.IsIntegral.of_finite
 
+/- warning: ring_hom.is_integral.to_finite -> RingHom.IsIntegral.to_finite is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] {f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))}, (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f) -> (RingHom.FiniteType.{u1, u2} R S _inst_1 _inst_4 f) -> (RingHom.Finite.{u1, u2} R S _inst_1 _inst_4 f)
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_4 : CommRing.{u1} S] {f : 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_4)))}, (RingHom.IsIntegral.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) f) -> (RingHom.FiniteType.{u2, u1} R S _inst_1 _inst_4 f) -> (RingHom.Finite.{u2, u1} R S _inst_1 _inst_4 f)
+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral.to_finite RingHom.IsIntegral.to_finiteₓ'. -/
 theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite :=
   by
   letI := f.to_algebra
@@ -425,17 +571,35 @@ theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.
   constructor
   change (⊤ : Subalgebra R S).toSubmodule.FG
   rw [← hs]
-  exact fG_adjoin_of_finite (Set.toFinite _) fun x _ => h x
+  exact FG_adjoin_of_finite (Set.toFinite _) fun x _ => h x
 #align ring_hom.is_integral.to_finite RingHom.IsIntegral.to_finite
 
+/- warning: ring_hom.finite.of_is_integral_of_finite_type -> RingHom.Finite.of_isIntegral_of_finiteType is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] {f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))}, (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f) -> (RingHom.FiniteType.{u1, u2} R S _inst_1 _inst_4 f) -> (RingHom.Finite.{u1, u2} R S _inst_1 _inst_4 f)
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_4 : CommRing.{u1} S] {f : 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_4)))}, (RingHom.IsIntegral.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) f) -> (RingHom.FiniteType.{u2, u1} R S _inst_1 _inst_4 f) -> (RingHom.Finite.{u2, u1} R S _inst_1 _inst_4 f)
+Case conversion may be inaccurate. Consider using '#align ring_hom.finite.of_is_integral_of_finite_type RingHom.Finite.of_isIntegral_of_finiteTypeₓ'. -/
 alias RingHom.IsIntegral.to_finite ← RingHom.Finite.of_isIntegral_of_finiteType
 #align ring_hom.finite.of_is_integral_of_finite_type RingHom.Finite.of_isIntegral_of_finiteType
 
+/- warning: ring_hom.finite_iff_is_integral_and_finite_type -> RingHom.finite_iff_isIntegral_and_finiteType is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] {f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))}, Iff (RingHom.Finite.{u1, u2} R S _inst_1 _inst_4 f) (And (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f) (RingHom.FiniteType.{u1, u2} R S _inst_1 _inst_4 f))
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_4 : CommRing.{u1} S] {f : 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_4)))}, Iff (RingHom.Finite.{u2, u1} R S _inst_1 _inst_4 f) (And (RingHom.IsIntegral.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) f) (RingHom.FiniteType.{u2, u1} R S _inst_1 _inst_4 f))
+Case conversion may be inaccurate. Consider using '#align ring_hom.finite_iff_is_integral_and_finite_type RingHom.finite_iff_isIntegral_and_finiteTypeₓ'. -/
 /-- finite = integral + finite type -/
 theorem RingHom.finite_iff_isIntegral_and_finiteType : f.Finite ↔ f.IsIntegral ∧ f.FiniteType :=
   ⟨fun h => ⟨h.to_isIntegral, h.to_finiteType⟩, fun ⟨h, h'⟩ => h.toFinite h'⟩
 #align ring_hom.finite_iff_is_integral_and_finite_type RingHom.finite_iff_isIntegral_and_finiteType
 
+/- warning: algebra.is_integral.finite -> Algebra.IsIntegral.finite is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))], (Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5) -> (forall [h' : Algebra.FiniteType.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5], Module.Finite.{u1, u2} R A (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))], (Algebra.IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5) -> (forall [h' : Algebra.FiniteType.{u1, u2} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5], Module.Finite.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5))
+Case conversion may be inaccurate. Consider using '#align algebra.is_integral.finite Algebra.IsIntegral.finiteₓ'. -/
 theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.FiniteType R A] :
     Module.Finite R A :=
   by
@@ -449,12 +613,24 @@ theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.Fin
   delta RingHom.Finite at this; convert this; ext; exact Algebra.smul_def _ _
 #align algebra.is_integral.finite Algebra.IsIntegral.finite
 
+/- warning: algebra.is_integral.of_finite -> Algebra.IsIntegral.of_finite is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [h : Module.Finite.{u1, u2} R A (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)], Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] [h : Module.Finite.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)], Algebra.IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5
+Case conversion may be inaccurate. Consider using '#align algebra.is_integral.of_finite Algebra.IsIntegral.of_finiteₓ'. -/
 theorem Algebra.IsIntegral.of_finite [h : Module.Finite R A] : Algebra.IsIntegral R A :=
   by
   apply RingHom.Finite.to_isIntegral
   delta RingHom.Finite; convert h; ext; exact (Algebra.smul_def _ _).symm
 #align algebra.is_integral.of_finite Algebra.IsIntegral.of_finite
 
+/- warning: algebra.finite_iff_is_integral_and_finite_type -> Algebra.finite_iff_isIntegral_and_finiteType is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))], Iff (Module.Finite.{u1, u2} R A (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (And (Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5) (Algebra.FiniteType.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))], Iff (Module.Finite.{u2, u1} R A (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (Algebra.toModule.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (And (Algebra.IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5) (Algebra.FiniteType.{u1, u2} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5))
+Case conversion may be inaccurate. Consider using '#align algebra.finite_iff_is_integral_and_finite_type Algebra.finite_iff_isIntegral_and_finiteTypeₓ'. -/
 /-- finite = integral + finite type -/
 theorem Algebra.finite_iff_isIntegral_and_finiteType :
     Module.Finite R A ↔ Algebra.IsIntegral R A ∧ Algebra.FiniteType R A :=
@@ -463,85 +639,177 @@ theorem Algebra.finite_iff_isIntegral_and_finiteType :
 
 variable (f)
 
+/- warning: ring_hom.is_integral_of_mem_closure -> RingHom.is_integral_of_mem_closure is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) {x : S} {y : S} {z : S}, (RingHom.IsIntegralElem.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f x) -> (RingHom.IsIntegralElem.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f y) -> (Membership.Mem.{u2, u2} S (Subring.{u2} S (CommRing.toRing.{u2} S _inst_4)) (SetLike.hasMem.{u2, u2} (Subring.{u2} S (CommRing.toRing.{u2} S _inst_4)) S (Subring.setLike.{u2} S (CommRing.toRing.{u2} S _inst_4))) z (Subring.closure.{u2} S (CommRing.toRing.{u2} S _inst_4) (Insert.insert.{u2, u2} S (Set.{u2} S) (Set.hasInsert.{u2} S) x (Singleton.singleton.{u2, u2} S (Set.{u2} S) (Set.hasSingleton.{u2} S) y)))) -> (RingHom.IsIntegralElem.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f z)
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_4 : CommRing.{u1} S] (f : 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_4)))) {x : S} {y : S} {z : S}, (RingHom.IsIntegralElem.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) f x) -> (RingHom.IsIntegralElem.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) f y) -> (Membership.mem.{u1, u1} S (Subring.{u1} S (CommRing.toRing.{u1} S _inst_4)) (SetLike.instMembership.{u1, u1} (Subring.{u1} S (CommRing.toRing.{u1} S _inst_4)) S (Subring.instSetLikeSubring.{u1} S (CommRing.toRing.{u1} S _inst_4))) z (Subring.closure.{u1} S (CommRing.toRing.{u1} S _inst_4) (Insert.insert.{u1, u1} S (Set.{u1} S) (Set.instInsertSet.{u1} S) x (Singleton.singleton.{u1, u1} S (Set.{u1} S) (Set.instSingletonSet.{u1} S) y)))) -> (RingHom.IsIntegralElem.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) f z)
+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_of_mem_closure RingHom.is_integral_of_mem_closureₓ'. -/
 theorem RingHom.is_integral_of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
     (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z :=
   by
   letI : Algebra R S := f.to_algebra
-  have := (fG_adjoin_singleton_of_integral x hx).mul (fG_adjoin_singleton_of_integral y hy)
+  have := (FG_adjoin_singleton_of_integral x hx).mul (FG_adjoin_singleton_of_integral y hy)
   rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this
   exact
-    isIntegral_of_mem_of_fG (Algebra.adjoin R {x, y}) this z
+    isIntegral_of_mem_of_FG (Algebra.adjoin R {x, y}) this z
       (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)
 #align ring_hom.is_integral_of_mem_closure RingHom.is_integral_of_mem_closure
 
+/- warning: is_integral_of_mem_closure -> isIntegral_of_mem_closure is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {x : A} {y : A} {z : A}, (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 y) -> (Membership.Mem.{u2, u2} A (Subring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (SetLike.hasMem.{u2, u2} (Subring.{u2} A (CommRing.toRing.{u2} A _inst_2)) A (Subring.setLike.{u2} A (CommRing.toRing.{u2} A _inst_2))) z (Subring.closure.{u2} A (CommRing.toRing.{u2} A _inst_2) (Insert.insert.{u2, u2} A (Set.{u2} A) (Set.hasInsert.{u2} A) x (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) y)))) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 z)
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {x : A} {y : A} {z : A}, (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 y) -> (Membership.mem.{u1, u1} A (Subring.{u1} A (CommRing.toRing.{u1} A _inst_2)) (SetLike.instMembership.{u1, u1} (Subring.{u1} A (CommRing.toRing.{u1} A _inst_2)) A (Subring.instSetLikeSubring.{u1} A (CommRing.toRing.{u1} A _inst_2))) z (Subring.closure.{u1} A (CommRing.toRing.{u1} A _inst_2) (Insert.insert.{u1, u1} A (Set.{u1} A) (Set.instInsertSet.{u1} A) x (Singleton.singleton.{u1, u1} A (Set.{u1} A) (Set.instSingletonSet.{u1} A) y)))) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 z)
+Case conversion may be inaccurate. Consider using '#align is_integral_of_mem_closure isIntegral_of_mem_closureₓ'. -/
 theorem isIntegral_of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
     (hz : z ∈ Subring.closure ({x, y} : Set A)) : IsIntegral R z :=
   (algebraMap R A).is_integral_of_mem_closure hx hy hz
 #align is_integral_of_mem_closure isIntegral_of_mem_closure
 
+/- warning: ring_hom.is_integral_zero -> RingHom.is_integral_zero is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))), RingHom.IsIntegralElem.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f (OfNat.ofNat.{u2} S 0 (OfNat.mk.{u2} S 0 (Zero.zero.{u2} S (MulZeroClass.toHasZero.{u2} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} S (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} S (NonAssocRing.toNonUnitalNonAssocRing.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))))))))
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_4 : CommRing.{u1} S] (f : 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_4)))), RingHom.IsIntegralElem.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) f (OfNat.ofNat.{u1} S 0 (Zero.toOfNat0.{u1} S (CommMonoidWithZero.toZero.{u1} S (CommSemiring.toCommMonoidWithZero.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)))))
+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_zero RingHom.is_integral_zeroₓ'. -/
 theorem RingHom.is_integral_zero : f.IsIntegralElem 0 :=
   f.map_zero ▸ f.is_integral_map
 #align ring_hom.is_integral_zero RingHom.is_integral_zero
 
+/- warning: is_integral_zero -> isIntegral_zero is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align is_integral_zero isIntegral_zeroₓ'. -/
 theorem isIntegral_zero : IsIntegral R (0 : A) :=
   (algebraMap R A).is_integral_zero
 #align is_integral_zero isIntegral_zero
 
+/- warning: ring_hom.is_integral_one -> RingHom.is_integral_one is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_one RingHom.is_integral_oneₓ'. -/
 theorem RingHom.is_integral_one : f.IsIntegralElem 1 :=
   f.map_one ▸ f.is_integral_map
 #align ring_hom.is_integral_one RingHom.is_integral_one
 
+/- warning: is_integral_one -> isIntegral_one is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_integral_one isIntegral_oneₓ'. -/
 theorem isIntegral_one : IsIntegral R (1 : A) :=
   (algebraMap R A).is_integral_one
 #align is_integral_one isIntegral_one
 
+/- warning: ring_hom.is_integral_add -> RingHom.is_integral_add is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_add RingHom.is_integral_addₓ'. -/
 theorem RingHom.is_integral_add {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x + y) :=
   f.is_integral_of_mem_closure hx hy <|
     Subring.add_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl))
 #align ring_hom.is_integral_add RingHom.is_integral_add
 
+/- warning: is_integral_add -> isIntegral_add is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_integral_add isIntegral_addₓ'. -/
 theorem isIntegral_add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x + y) :=
   (algebraMap R A).is_integral_add hx hy
 #align is_integral_add isIntegral_add
 
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+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_neg RingHom.is_integral_negₓ'. -/
 theorem RingHom.is_integral_neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) :=
   f.is_integral_of_mem_closure hx hx (Subring.neg_mem _ (Subring.subset_closure (Or.inl rfl)))
 #align ring_hom.is_integral_neg RingHom.is_integral_neg
 
+/- warning: is_integral_neg -> isIntegral_neg is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_integral_neg isIntegral_negₓ'. -/
 theorem isIntegral_neg {x : A} (hx : IsIntegral R x) : IsIntegral R (-x) :=
   (algebraMap R A).is_integral_neg hx
 #align is_integral_neg isIntegral_neg
 
+/- warning: ring_hom.is_integral_sub -> RingHom.is_integral_sub is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_sub RingHom.is_integral_subₓ'. -/
 theorem RingHom.is_integral_sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x - y) := by
   simpa only [sub_eq_add_neg] using f.is_integral_add hx (f.is_integral_neg hy)
 #align ring_hom.is_integral_sub RingHom.is_integral_sub
 
+/- warning: is_integral_sub -> isIntegral_sub is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_integral_sub isIntegral_subₓ'. -/
 theorem isIntegral_sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x - y) :=
   (algebraMap R A).is_integral_sub hx hy
 #align is_integral_sub isIntegral_sub
 
+/- warning: ring_hom.is_integral_mul -> RingHom.is_integral_mul is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_mul RingHom.is_integral_mulₓ'. -/
 theorem RingHom.is_integral_mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x * y) :=
   f.is_integral_of_mem_closure hx hy
     (Subring.mul_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl)))
 #align ring_hom.is_integral_mul RingHom.is_integral_mul
 
+/- warning: is_integral_mul -> isIntegral_mul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {x : A} {y : A}, (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 y) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (HMul.hMul.{u2, u2, u2} A A A (instHMul.{u2} A (Distrib.toHasMul.{u2} A (Ring.toDistrib.{u2} A (CommRing.toRing.{u2} A _inst_2)))) x y))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {x : A} {y : A}, (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 y) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (HMul.hMul.{u1, u1, u1} A A A (instHMul.{u1} A (NonUnitalNonAssocRing.toMul.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) x y))
+Case conversion may be inaccurate. Consider using '#align is_integral_mul isIntegral_mulₓ'. -/
 theorem isIntegral_mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x * y) :=
   (algebraMap R A).is_integral_mul hx hy
 #align is_integral_mul isIntegral_mul
 
+#print isIntegral_smul /-
 theorem isIntegral_smul [Algebra S A] [Algebra R S] [IsScalarTower R S A] {x : A} (r : R)
     (hx : IsIntegral S x) : IsIntegral S (r • x) :=
   by
   rw [Algebra.smul_def, IsScalarTower.algebraMap_apply R S A]
   exact isIntegral_mul isIntegral_algebraMap hx
 #align is_integral_smul isIntegral_smul
+-/
 
+/- warning: is_integral_of_pow -> isIntegral_of_pow is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {x : A} {n : Nat}, (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) n) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (Ring.toMonoid.{u2} A (CommRing.toRing.{u2} A _inst_2)))) x n)) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x)
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {x : A} {n : Nat}, (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (HPow.hPow.{u1, 0, u1} A Nat A (instHPow.{u1, 0} A Nat (Monoid.Pow.{u1} A (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))))) x n)) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x)
+Case conversion may be inaccurate. Consider using '#align is_integral_of_pow isIntegral_of_powₓ'. -/
 theorem isIntegral_of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
     IsIntegral R x := by
   rcases hx with ⟨p, ⟨hmonic, heval⟩⟩
@@ -552,6 +820,7 @@ theorem isIntegral_of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x
 
 variable (R A)
 
+#print integralClosure /-
 /-- The integral closure of R in an R-algebra A. -/
 def integralClosure : Subalgebra R A
     where
@@ -562,16 +831,29 @@ def integralClosure : Subalgebra R A
   mul_mem' _ _ := isIntegral_mul
   algebraMap_mem' x := isIntegral_algebraMap
 #align integral_closure integralClosure
+-/
 
-theorem mem_integralClosure_iff_mem_fG {r : A} :
+/- warning: mem_integral_closure_iff_mem_fg -> mem_integralClosure_iff_mem_FG is a dubious translation:
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(CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (LE.le.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) 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(Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteSemilatticeInf.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (CompleteLattice.toCompleteSemilatticeInf.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)))))))) (Subalgebra.toSubmodule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) M)) (Membership.Mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) r M)))
+but is expected to have type
+  forall (R : Type.{u1}) (A : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] {r : A}, Iff (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) r (integralClosure.{u1, u2} R A _inst_1 _inst_2 _inst_5)) (Exists.{succ u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (fun (M : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) => And (Submodule.FG.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (FunLike.coe.{succ u2, succ u2, succ u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5))))))) (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (fun (_x : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) => (fun (x._@.Mathlib.Order.RelIso.Basic._hyg.869 : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) => Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) _x) (RelHomClass.toFunLike.{u2, u2, u2} (OrderEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)))) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5))))))) (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) => LE.le.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)))) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699) (RelEmbedding.instRelHomClassRelEmbedding.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.682 : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (x._@.Mathlib.Order.Hom.Basic._hyg.684 : Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) => LE.le.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (Preorder.toLE.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (SetLike.instPartialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)))) x._@.Mathlib.Order.Hom.Basic._hyg.682 x._@.Mathlib.Order.Hom.Basic._hyg.684) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.697 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (x._@.Mathlib.Order.Hom.Basic._hyg.699 : Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) => LE.le.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Preorder.toLE.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) (Submodule.completeLattice.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)))))) x._@.Mathlib.Order.Hom.Basic._hyg.697 x._@.Mathlib.Order.Hom.Basic._hyg.699))) (Subalgebra.toSubmodule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) M)) (Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_5)) r M)))
+Case conversion may be inaccurate. Consider using '#align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_FGₓ'. -/
+theorem mem_integralClosure_iff_mem_FG {r : A} :
     r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M :=
   ⟨fun hr =>
-    ⟨Algebra.adjoin R {r}, fG_adjoin_singleton_of_integral _ hr, Algebra.subset_adjoin rfl⟩,
-    fun ⟨M, Hf, hrM⟩ => isIntegral_of_mem_of_fG M Hf _ hrM⟩
-#align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_fG
+    ⟨Algebra.adjoin R {r}, FG_adjoin_singleton_of_integral _ hr, Algebra.subset_adjoin rfl⟩,
+    fun ⟨M, Hf, hrM⟩ => isIntegral_of_mem_of_FG M Hf _ hrM⟩
+#align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_FG
 
 variable {R} {A}
 
+/- warning: adjoin_le_integral_closure -> adjoin_le_integralClosure is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {x : A}, (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x) -> (LE.le.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Preorder.toHasLe.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (SetLike.partialOrder.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)))) (Algebra.adjoin.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5 (Singleton.singleton.{u2, u2} A (Set.{u2} A) (Set.hasSingleton.{u2} A) x)) (integralClosure.{u1, u2} R A _inst_1 _inst_2 _inst_5))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {x : A}, (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x) -> (LE.le.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Preorder.toLE.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (PartialOrder.toPreorder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.instCompleteLatticeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5))))) (Algebra.adjoin.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5 (Singleton.singleton.{u1, u1} A (Set.{u1} A) (Set.instSingletonSet.{u1} A) x)) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_5))
+Case conversion may be inaccurate. Consider using '#align adjoin_le_integral_closure adjoin_le_integralClosureₓ'. -/
 theorem adjoin_le_integralClosure {x : A} (hx : IsIntegral R x) :
     Algebra.adjoin R {x} ≤ integralClosure R A :=
   by
@@ -580,6 +862,12 @@ theorem adjoin_le_integralClosure {x : A} (hx : IsIntegral R x) :
   exact hx
 #align adjoin_le_integral_closure adjoin_le_integralClosure
 
+/- warning: le_integral_closure_iff_is_integral -> le_integralClosure_iff_isIntegral is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align le_integral_closure_iff_is_integral le_integralClosure_iff_isIntegralₓ'. -/
 theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
     S ≤ integralClosure R A ↔ Algebra.IsIntegral R S :=
   SetLike.forall.symm.trans
@@ -588,11 +876,23 @@ theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
         isIntegral_algebraMap_iff Subtype.coe_injective)
 #align le_integral_closure_iff_is_integral le_integralClosure_iff_isIntegral
 
+/- warning: is_integral_sup -> isIntegral_sup is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_integral_sup isIntegral_supₓ'. -/
 theorem isIntegral_sup {S T : Subalgebra R A} :
     Algebra.IsIntegral R ↥(S ⊔ T) ↔ Algebra.IsIntegral R S ∧ Algebra.IsIntegral R T := by
   simp only [← le_integralClosure_iff_isIntegral, sup_le_iff]
 #align is_integral_sup isIntegral_sup
 
+/- warning: integral_closure_map_alg_equiv -> integralClosure_map_algEquiv is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align integral_closure_map_alg_equiv integralClosure_map_algEquivₓ'. -/
 /-- Mapping an integral closure along an `alg_equiv` gives the integral closure. -/
 theorem integralClosure_map_algEquiv (f : A ≃ₐ[R] B) :
     (integralClosure R A).map (f : A →ₐ[R] B) = integralClosure R B :=
@@ -607,6 +907,12 @@ theorem integralClosure_map_algEquiv (f : A ≃ₐ[R] B) :
     simp
 #align integral_closure_map_alg_equiv integralClosure_map_algEquiv
 
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+Case conversion may be inaccurate. Consider using '#align integral_closure.is_integral integralClosure.isIntegralₓ'. -/
 theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
   let ⟨p, hpm, hpx⟩ := x.2
   ⟨p, hpm,
@@ -614,6 +920,12 @@ theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
       rwa [← aeval_def, Subtype.val_eq_coe, ← Subalgebra.val_apply, aeval_alg_hom_apply] at hpx⟩
 #align integral_closure.is_integral integralClosure.isIntegral
 
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+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_of_is_integral_mul_unit RingHom.is_integral_of_is_integral_mul_unitₓ'. -/
 theorem RingHom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
     (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x :=
   by
@@ -623,11 +935,23 @@ theorem RingHom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r
   rw [mul_comm x y, ← mul_assoc, hr, one_mul]
 #align ring_hom.is_integral_of_is_integral_mul_unit RingHom.is_integral_of_is_integral_mul_unit
 
+/- warning: is_integral_of_is_integral_mul_unit -> isIntegral_of_isIntegral_mul_unit is a dubious translation:
+lean 3 declaration is
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+Case conversion may be inaccurate. Consider using '#align is_integral_of_is_integral_mul_unit isIntegral_of_isIntegral_mul_unitₓ'. -/
 theorem isIntegral_of_isIntegral_mul_unit {x y : A} {r : R} (hr : algebraMap R A r * y = 1)
     (hx : IsIntegral R (x * y)) : IsIntegral R x :=
   (algebraMap R A).is_integral_of_is_integral_mul_unit x y r hr hx
 #align is_integral_of_is_integral_mul_unit isIntegral_of_isIntegral_mul_unit
 
+/- warning: is_integral_of_mem_closure' -> isIntegral_of_mem_closure' is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] (G : Set.{u2} A), (forall (x : A), (Membership.Mem.{u2, u2} A (Set.{u2} A) (Set.hasMem.{u2} A) x G) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x)) -> (forall (x : A), (Membership.Mem.{u2, u2} A (Subring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (SetLike.hasMem.{u2, u2} (Subring.{u2} A (CommRing.toRing.{u2} A _inst_2)) A (Subring.setLike.{u2} A (CommRing.toRing.{u2} A _inst_2))) x (Subring.closure.{u2} A (CommRing.toRing.{u2} A _inst_2) G)) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] (G : Set.{u2} A), (forall (x : A), (Membership.mem.{u2, u2} A (Set.{u2} A) (Set.instMembershipSet.{u2} A) x G) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x)) -> (forall (x : A), (Membership.mem.{u2, u2} A (Subring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (SetLike.instMembership.{u2, u2} (Subring.{u2} A (CommRing.toRing.{u2} A _inst_2)) A (Subring.instSetLikeSubring.{u2} A (CommRing.toRing.{u2} A _inst_2))) x (Subring.closure.{u2} A (CommRing.toRing.{u2} A _inst_2) G)) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x))
+Case conversion may be inaccurate. Consider using '#align is_integral_of_mem_closure' isIntegral_of_mem_closure'ₓ'. -/
 /-- Generalization of `is_integral_of_mem_closure` bootstrapped up from that lemma -/
 theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) :
     ∀ x ∈ Subring.closure G, IsIntegral R x := fun x hx =>
@@ -635,50 +959,98 @@ theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x
     (fun _ => isIntegral_neg) fun _ _ => isIntegral_mul
 #align is_integral_of_mem_closure' isIntegral_of_mem_closure'
 
+/- warning: is_integral_of_mem_closure'' -> is_integral_of_mem_closure'' is a dubious translation:
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+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {S : Type.{u2}} [_inst_7 : CommRing.{u2} S] {f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_7)))} (G : Set.{u2} S), (forall (x : S), (Membership.Mem.{u2, u2} S (Set.{u2} S) (Set.hasMem.{u2} S) x G) -> (RingHom.IsIntegralElem.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_7) f x)) -> (forall (x : S), (Membership.Mem.{u2, u2} S (Subring.{u2} S (CommRing.toRing.{u2} S _inst_7)) (SetLike.hasMem.{u2, u2} (Subring.{u2} S (CommRing.toRing.{u2} S _inst_7)) S (Subring.setLike.{u2} S (CommRing.toRing.{u2} S _inst_7))) x (Subring.closure.{u2} S (CommRing.toRing.{u2} S _inst_7) G)) -> (RingHom.IsIntegralElem.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_7) f x))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] {S : Type.{u2}} [_inst_7 : CommRing.{u2} S] {f : 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_7)))} (G : Set.{u2} S), (forall (x : S), (Membership.mem.{u2, u2} S (Set.{u2} S) (Set.instMembershipSet.{u2} S) x G) -> (RingHom.IsIntegralElem.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_7) f x)) -> (forall (x : S), (Membership.mem.{u2, u2} S (Subring.{u2} S (CommRing.toRing.{u2} S _inst_7)) (SetLike.instMembership.{u2, u2} (Subring.{u2} S (CommRing.toRing.{u2} S _inst_7)) S (Subring.instSetLikeSubring.{u2} S (CommRing.toRing.{u2} S _inst_7))) x (Subring.closure.{u2} S (CommRing.toRing.{u2} S _inst_7) G)) -> (RingHom.IsIntegralElem.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_7) f x))
+Case conversion may be inaccurate. Consider using '#align is_integral_of_mem_closure'' is_integral_of_mem_closure''ₓ'. -/
 theorem is_integral_of_mem_closure'' {S : Type _} [CommRing S] {f : R →+* S} (G : Set S)
     (hG : ∀ x ∈ G, f.IsIntegralElem x) : ∀ x ∈ Subring.closure G, f.IsIntegralElem x := fun x hx =>
   @isIntegral_of_mem_closure' R S _ _ f.toAlgebra G hG x hx
 #align is_integral_of_mem_closure'' is_integral_of_mem_closure''
 
+/- warning: is_integral.pow -> IsIntegral.pow is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {x : A}, (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x) -> (forall (n : Nat), IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (Ring.toMonoid.{u2} A (CommRing.toRing.{u2} A _inst_2)))) x n))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {x : A}, (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x) -> (forall (n : Nat), IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (HPow.hPow.{u1, 0, u1} A Nat A (instHPow.{u1, 0} A Nat (Monoid.Pow.{u1} A (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))))) x n))
+Case conversion may be inaccurate. Consider using '#align is_integral.pow IsIntegral.powₓ'. -/
 theorem IsIntegral.pow {x : A} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (x ^ n) :=
   (integralClosure R A).pow_mem h n
 #align is_integral.pow IsIntegral.pow
 
+/- warning: is_integral.nsmul -> IsIntegral.nsmul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {x : A}, (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x) -> (forall (n : Nat), IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (SMul.smul.{0, u2} Nat A (AddMonoid.SMul.{u2} A (AddMonoidWithOne.toAddMonoid.{u2} A (AddGroupWithOne.toAddMonoidWithOne.{u2} A (AddCommGroupWithOne.toAddGroupWithOne.{u2} A (Ring.toAddCommGroupWithOne.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) n x))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {x : A}, (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x) -> (forall (n : Nat), IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (HSMul.hSMul.{0, u1, u1} Nat A A (instHSMul.{0, u1} Nat A (AddMonoid.SMul.{u1} A (AddMonoidWithOne.toAddMonoid.{u1} A (AddGroupWithOne.toAddMonoidWithOne.{u1} A (Ring.toAddGroupWithOne.{u1} A (CommRing.toRing.{u1} A _inst_2)))))) n x))
+Case conversion may be inaccurate. Consider using '#align is_integral.nsmul IsIntegral.nsmulₓ'. -/
 theorem IsIntegral.nsmul {x : A} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (n • x) :=
   (integralClosure R A).nsmul_mem h n
 #align is_integral.nsmul IsIntegral.nsmul
 
+/- warning: is_integral.zsmul -> IsIntegral.zsmul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {x : A}, (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x) -> (forall (n : Int), IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (SMul.smul.{0, u2} Int A (SubNegMonoid.SMulInt.{u2} A (AddGroup.toSubNegMonoid.{u2} A (AddGroupWithOne.toAddGroup.{u2} A (AddCommGroupWithOne.toAddGroupWithOne.{u2} A (Ring.toAddCommGroupWithOne.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) n x))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {x : A}, (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x) -> (forall (n : Int), IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (HSMul.hSMul.{0, u1, u1} Int A A (instHSMul.{0, u1} Int A (SubNegMonoid.SMulInt.{u1} A (AddGroup.toSubNegMonoid.{u1} A (AddGroupWithOne.toAddGroup.{u1} A (Ring.toAddGroupWithOne.{u1} A (CommRing.toRing.{u1} A _inst_2)))))) n x))
+Case conversion may be inaccurate. Consider using '#align is_integral.zsmul IsIntegral.zsmulₓ'. -/
 theorem IsIntegral.zsmul {x : A} (h : IsIntegral R x) (n : ℤ) : IsIntegral R (n • x) :=
   (integralClosure R A).zsmul_mem h n
 #align is_integral.zsmul IsIntegral.zsmul
 
+#print IsIntegral.multiset_prod /-
 theorem IsIntegral.multiset_prod {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) :
     IsIntegral R s.Prod :=
   (integralClosure R A).multiset_prod_mem h
 #align is_integral.multiset_prod IsIntegral.multiset_prod
+-/
 
+#print IsIntegral.multiset_sum /-
 theorem IsIntegral.multiset_sum {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) :
     IsIntegral R s.Sum :=
   (integralClosure R A).multiset_sum_mem h
 #align is_integral.multiset_sum IsIntegral.multiset_sum
+-/
 
+/- warning: is_integral.prod -> IsIntegral.prod is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {α : Type.{u3}} {s : Finset.{u3} α} (f : α -> A), (forall (x : α), (Membership.Mem.{u3, u3} α (Finset.{u3} α) (Finset.hasMem.{u3} α) x s) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (f x))) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (Finset.prod.{u2, u3} A α (CommRing.toCommMonoid.{u2} A _inst_2) s (fun (x : α) => f x)))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {α : Type.{u3}} {s : Finset.{u3} α} (f : α -> A), (forall (x : α), (Membership.mem.{u3, u3} α (Finset.{u3} α) (Finset.instMembershipFinset.{u3} α) x s) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (f x))) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (Finset.prod.{u1, u3} A α (CommRing.toCommMonoid.{u1} A _inst_2) s (fun (x : α) => f x)))
+Case conversion may be inaccurate. Consider using '#align is_integral.prod IsIntegral.prodₓ'. -/
 theorem IsIntegral.prod {α : Type _} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
     IsIntegral R (∏ x in s, f x) :=
   (integralClosure R A).prod_mem h
 #align is_integral.prod IsIntegral.prod
 
+/- warning: is_integral.sum -> IsIntegral.sum is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {α : Type.{u3}} {s : Finset.{u3} α} (f : α -> A), (forall (x : α), (Membership.Mem.{u3, u3} α (Finset.{u3} α) (Finset.hasMem.{u3} α) x s) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (f x))) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (Finset.sum.{u2, u3} A α (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) s (fun (x : α) => f x)))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {α : Type.{u3}} {s : Finset.{u3} α} (f : α -> A), (forall (x : α), (Membership.mem.{u3, u3} α (Finset.{u3} α) (Finset.instMembershipFinset.{u3} α) x s) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (f x))) -> (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (Finset.sum.{u1, u3} A α (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) s (fun (x : α) => f x)))
+Case conversion may be inaccurate. Consider using '#align is_integral.sum IsIntegral.sumₓ'. -/
 theorem IsIntegral.sum {α : Type _} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
     IsIntegral R (∑ x in s, f x) :=
   (integralClosure R A).sum_mem h
 #align is_integral.sum IsIntegral.sum
 
+#print IsIntegral.det /-
 theorem IsIntegral.det {n : Type _} [Fintype n] [DecidableEq n] {M : Matrix n n A}
     (h : ∀ i j, IsIntegral R (M i j)) : IsIntegral R M.det :=
   by
   rw [Matrix.det_apply]
   exact IsIntegral.sum _ fun σ hσ => IsIntegral.zsmul (IsIntegral.prod _ fun i hi => h _ _) _
 #align is_integral.det IsIntegral.det
+-/
 
+/- warning: is_integral.pow_iff -> IsIntegral.pow_iff is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {x : A} {n : Nat}, (LT.lt.{0} Nat Nat.hasLt (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero))) n) -> (Iff (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 (HPow.hPow.{u2, 0, u2} A Nat A (instHPow.{u2, 0} A Nat (Monoid.Pow.{u2} A (Ring.toMonoid.{u2} A (CommRing.toRing.{u2} A _inst_2)))) x n)) (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_5 x))
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_5 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] {x : A} {n : Nat}, (LT.lt.{0} Nat instLTNat (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0)) n) -> (Iff (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 (HPow.hPow.{u1, 0, u1} A Nat A (instHPow.{u1, 0} A Nat (Monoid.Pow.{u1} A (MonoidWithZero.toMonoid.{u1} A (Semiring.toMonoidWithZero.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))))) x n)) (IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_5 x))
+Case conversion may be inaccurate. Consider using '#align is_integral.pow_iff IsIntegral.pow_iffₓ'. -/
 @[simp]
 theorem IsIntegral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n) ↔ IsIntegral R x :=
   ⟨isIntegral_of_pow hn, fun hx => IsIntegral.pow hx n⟩
@@ -686,6 +1058,12 @@ theorem IsIntegral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n)
 
 open TensorProduct
 
+/- warning: is_integral.tmul -> IsIntegral.tmul is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_5 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_6 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] (x : A) {y : B}, (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_6 y) -> (IsIntegral.{u2, max u2 u3} A (TensorProduct.{u1, u2, u3} R (CommRing.toCommSemiring.{u1} R _inst_1) A B (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (AddCommGroup.toAddCommMonoid.{u3} B (NonUnitalNonAssocRing.toAddCommGroup.{u3} B (NonAssocRing.toNonUnitalNonAssocRing.{u3} B (Ring.toNonAssocRing.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6)) _inst_2 (Algebra.TensorProduct.TensorProduct.ring.{u1, u2, u3} R _inst_1 A (CommRing.toRing.{u2} A _inst_2) _inst_5 B (CommRing.toRing.{u3} B _inst_3) _inst_6) (Algebra.TensorProduct.leftAlgebra.{u1, u2, u3, u2} R (CommRing.toCommSemiring.{u1} R _inst_1) A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5 B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6 A (CommRing.toCommSemiring.{u2} A _inst_2) _inst_5 (Algebra.id.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (IsScalarTower.right.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5)) (TensorProduct.tmul.{u1, u2, u3} R (CommRing.toCommSemiring.{u1} R _inst_1) A B (AddCommGroup.toAddCommMonoid.{u2} A (NonUnitalNonAssocRing.toAddCommGroup.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (AddCommGroup.toAddCommMonoid.{u3} B (NonUnitalNonAssocRing.toAddCommGroup.{u3} B (NonAssocRing.toNonUnitalNonAssocRing.{u3} B (Ring.toNonAssocRing.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_5) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6) x y))
+but is expected to have type
+  forall {R : Type.{u3}} {A : Type.{u1}} {B : Type.{u2}} [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : CommRing.{u2} B] [_inst_5 : Algebra.{u3, u1} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] [_inst_6 : Algebra.{u3, u2} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] (x : A) {y : B}, (IsIntegral.{u3, u2} R B _inst_1 (CommRing.toRing.{u2} B _inst_3) _inst_6 y) -> (IsIntegral.{u1, max u2 u1} A (TensorProduct.{u3, u1, u2} R (CommRing.toCommSemiring.{u3} R _inst_1) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} B (NonAssocRing.toNonUnitalNonAssocRing.{u2} B (Ring.toNonAssocRing.{u2} B (CommRing.toRing.{u2} B _inst_3))))) (Algebra.toModule.{u3, u1} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.toModule.{u3, u2} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)) _inst_2 (Algebra.TensorProduct.instRingTensorProductToCommSemiringToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToAddCommMonoidToNonUnitalNonAssocSemiringToNonUnitalNonAssocRingToNonAssocRingToModuleToSemiringToModuleToSemiring.{u3, u1, u2} R _inst_1 A (CommRing.toRing.{u1} A _inst_2) _inst_5 B (CommRing.toRing.{u2} B _inst_3) _inst_6) (Algebra.TensorProduct.leftAlgebra.{u3, u1, u2, u1} R (CommRing.toCommSemiring.{u3} R _inst_1) A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5 B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6 A (CommRing.toCommSemiring.{u1} A _inst_2) _inst_5 (Algebra.id.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (IsScalarTower.right.{u3, u1} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5)) (TensorProduct.tmul.{u3, u1, u2} R (CommRing.toCommSemiring.{u3} R _inst_1) A B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} B (NonAssocRing.toNonUnitalNonAssocRing.{u2} B (Ring.toNonAssocRing.{u2} B (CommRing.toRing.{u2} B _inst_3))))) (Algebra.toModule.{u3, u1} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_5) (Algebra.toModule.{u3, u2} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) x y))
+Case conversion may be inaccurate. Consider using '#align is_integral.tmul IsIntegral.tmulₓ'. -/
 theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x ⊗ₜ[R] y) :=
   by
   obtain ⟨p, hp, hp'⟩ := h
@@ -709,12 +1087,20 @@ section
 
 variable (p : R[X]) (x : S)
 
+#print normalizeScaleRoots /-
 /-- The monic polynomial whose roots are `p.leading_coeff * x` for roots `x` of `p`. -/
 noncomputable def normalizeScaleRoots (p : R[X]) : R[X] :=
   ∑ i in p.support,
     monomial i (if i = p.natDegree then 1 else p.coeff i * p.leadingCoeff ^ (p.natDegree - 1 - i))
 #align normalize_scale_roots normalizeScaleRoots
+-/
 
+/- warning: normalize_scale_roots_coeff_mul_leading_coeff_pow -> normalizeScaleRoots_coeff_mul_leadingCoeff_pow is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (p : Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (i : Nat), (LE.le.{0} Nat Nat.hasLe (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) (Polynomial.natDegree.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) p)) -> (Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (normalizeScaleRoots.{u1} R _inst_1 p) i) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Polynomial.leadingCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) i)) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (Ring.toDistrib.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Polynomial.coeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) p i) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R _inst_1)))) (Polynomial.leadingCoeff.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Polynomial.natDegree.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) p) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))))
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (i : Nat), (LE.le.{0} Nat instLENat (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) p)) -> (Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (normalizeScaleRoots.{u1} R _inst_1 p) i) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) p) i)) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Polynomial.coeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) p i) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))))) (Polynomial.leadingCoeff.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) p) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Polynomial.natDegree.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) p) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))))
+Case conversion may be inaccurate. Consider using '#align normalize_scale_roots_coeff_mul_leading_coeff_pow normalizeScaleRoots_coeff_mul_leadingCoeff_powₓ'. -/
 theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ natDegree p) :
     (normalizeScaleRoots p).coeff i * p.leadingCoeff ^ i =
       p.coeff i * p.leadingCoeff ^ (p.natDegree - 1) :=
@@ -730,6 +1116,7 @@ theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ nat
     exact ⟨le_nat_degree_of_ne_zero h₁, h₂⟩
 #align normalize_scale_roots_coeff_mul_leading_coeff_pow normalizeScaleRoots_coeff_mul_leadingCoeff_pow
 
+#print leadingCoeff_smul_normalizeScaleRoots /-
 theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
     p.leadingCoeff • normalizeScaleRoots p = scaleRoots p p.leadingCoeff :=
   by
@@ -744,7 +1131,14 @@ theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
     rw [Nat.succ_le_iff]
     exact tsub_pos_of_lt (lt_of_le_of_ne (le_nat_degree_of_ne_zero h₁) h₂)
 #align leading_coeff_smul_normalize_scale_roots leadingCoeff_smul_normalizeScaleRoots
+-/
 
+/- warning: normalize_scale_roots_support -> normalizeScaleRoots_support is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (p : Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))), LE.le.{0} (Finset.{0} Nat) (Preorder.toHasLe.{0} (Finset.{0} Nat) (PartialOrder.toPreorder.{0} (Finset.{0} Nat) (Finset.partialOrder.{0} Nat))) (Polynomial.support.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (normalizeScaleRoots.{u1} R _inst_1 p)) (Polynomial.support.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) p)
+but is expected to have type
+  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (p : Polynomial.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))), LE.le.{0} (Finset.{0} Nat) (Preorder.toLE.{0} (Finset.{0} Nat) (PartialOrder.toPreorder.{0} (Finset.{0} Nat) (Finset.partialOrder.{0} Nat))) (Polynomial.support.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (normalizeScaleRoots.{u1} R _inst_1 p)) (Polynomial.support.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) p)
+Case conversion may be inaccurate. Consider using '#align normalize_scale_roots_support normalizeScaleRoots_supportₓ'. -/
 theorem normalizeScaleRoots_support : (normalizeScaleRoots p).support ≤ p.support :=
   by
   intro x
@@ -755,6 +1149,7 @@ theorem normalizeScaleRoots_support : (normalizeScaleRoots p).support ≤ p.supp
   exact (h₂ h₁).rec _
 #align normalize_scale_roots_support normalizeScaleRoots_support
 
+#print normalizeScaleRoots_degree /-
 theorem normalizeScaleRoots_degree : (normalizeScaleRoots p).degree = p.degree :=
   by
   apply le_antisymm
@@ -762,7 +1157,14 @@ theorem normalizeScaleRoots_degree : (normalizeScaleRoots p).degree = p.degree :
   · rw [← degree_scale_roots, ← leadingCoeff_smul_normalizeScaleRoots]
     exact degree_smul_le _ _
 #align normalize_scale_roots_degree normalizeScaleRoots_degree
+-/
 
+/- warning: normalize_scale_roots_eval₂_leading_coeff_mul -> normalizeScaleRoots_eval₂_leadingCoeff_mul is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align normalize_scale_roots_eval₂_leading_coeff_mul normalizeScaleRoots_eval₂_leadingCoeff_mulₓ'. -/
 theorem normalizeScaleRoots_eval₂_leadingCoeff_mul (h : 1 ≤ p.natDegree) (f : R →+* S) (x : S) :
     (normalizeScaleRoots p).eval₂ f (f p.leadingCoeff * x) =
       f p.leadingCoeff ^ (p.natDegree - 1) * p.eval₂ f x :=
@@ -776,6 +1178,7 @@ theorem normalizeScaleRoots_eval₂_leadingCoeff_mul (h : 1 ≤ p.natDegree) (f
   ring
 #align normalize_scale_roots_eval₂_leading_coeff_mul normalizeScaleRoots_eval₂_leadingCoeff_mul
 
+#print normalizeScaleRoots_monic /-
 theorem normalizeScaleRoots_monic (h : p ≠ 0) : (normalizeScaleRoots p).Monic :=
   by
   delta monic leading_coeff
@@ -783,7 +1186,14 @@ theorem normalizeScaleRoots_monic (h : p ≠ 0) : (normalizeScaleRoots p).Monic
   suffices p = 0 → (0 : R) = 1 by simpa [normalizeScaleRoots, coeff_monomial]
   exact fun h' => (h h').rec _
 #align normalize_scale_roots_monic normalizeScaleRoots_monic
+-/
 
+/- warning: ring_hom.is_integral_elem_leading_coeff_mul -> RingHom.isIntegralElem_leadingCoeff_mul is a dubious translation:
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_elem_leading_coeff_mul RingHom.isIntegralElem_leadingCoeff_mulₓ'. -/
 /-- Given a `p : R[X]` and a `x : S` such that `p.eval₂ f x = 0`,
 `f p.leading_coeff * x` is integral. -/
 theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
@@ -807,6 +1217,12 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
       rw [eq_C_of_nat_degree_eq_zero h', map_C, h, C_eq_zero]
 #align ring_hom.is_integral_elem_leading_coeff_mul RingHom.isIntegralElem_leadingCoeff_mul
 
+/- warning: is_integral_leading_coeff_smul -> isIntegral_leadingCoeff_smul is a dubious translation:
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_inst_1))) S (CommRing.toCommSemiring.{u2} R _inst_1) (Polynomial.semiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) (Polynomial.algebraOfAlgebra.{u2, u2} R R (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (Algebra.id.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) _inst_7))))) (Polynomial.aeval.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7 x) p) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommMonoidWithZero.toZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommSemiring.toCommMonoidWithZero.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) (CommRing.toCommSemiring.{u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) => S) p) _inst_4)))))) -> (IsIntegral.{u2, u1} R S _inst_1 (CommRing.toRing.{u1} S _inst_4) _inst_7 (HSMul.hSMul.{u2, u1, u1} R S S (instHSMul.{u2, u1} R S (Algebra.toSMul.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)) _inst_7)) (Polynomial.leadingCoeff.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) p) x))
+Case conversion may be inaccurate. Consider using '#align is_integral_leading_coeff_smul isIntegral_leadingCoeff_smulₓ'. -/
 /-- Given a `p : R[X]` and a root `x : S`,
 then `p.leading_coeff • x : S` is integral over `R`. -/
 theorem isIntegral_leadingCoeff_smul [Algebra R S] (h : aeval x p = 0) :
@@ -823,6 +1239,7 @@ end
 
 section IsIntegralClosure
 
+#print IsIntegralClosure /-
 /- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
 /-- `is_integral_closure A R B` is the characteristic predicate stating `A` is
 the integral closure of `R` in `B`,
@@ -833,7 +1250,14 @@ class IsIntegralClosure (A R B : Type _) [CommRing R] [CommSemiring A] [CommRing
   algebraMap_injective : Function.Injective (algebraMap A B)
   isIntegral_iff : ∀ {x : B}, IsIntegral R x ↔ ∃ y, algebraMap A B y = x
 #align is_integral_closure IsIntegralClosure
+-/
 
+/- warning: integral_closure.is_integral_closure -> integralClosure.isIntegralClosure is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) (A : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))], IsIntegralClosure.{u2, u1, u2} (coeSort.{succ u2, succ (succ u2)} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3) A (Subalgebra.setLike.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3)) (integralClosure.{u1, u2} R A _inst_1 _inst_2 _inst_3)) R A _inst_1 (Subalgebra.toCommSemiring.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 (integralClosure.{u1, u2} R A _inst_1 _inst_2 _inst_3)) _inst_2 _inst_3 (Subalgebra.toAlgebra.{u2, u1, u2} A R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_3 (Algebra.id.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (integralClosure.{u1, u2} R A _inst_1 _inst_2 _inst_3))
+but is expected to have type
+  forall (R : Type.{u1}) (A : Type.{u2}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))], IsIntegralClosure.{u2, u1, u2} (Subtype.{succ u2} A (fun (x : A) => Membership.mem.{u2, u2} A (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3)) x (integralClosure.{u1, u2} R A _inst_1 _inst_2 _inst_3))) R A _inst_1 (Subalgebra.toCommSemiring.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 (integralClosure.{u1, u2} R A _inst_1 _inst_2 _inst_3)) _inst_2 _inst_3 (Subalgebra.toAlgebra.{u2, u1, u2} A R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_3 (Algebra.id.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (integralClosure.{u1, u2} R A _inst_1 _inst_2 _inst_3))
+Case conversion may be inaccurate. Consider using '#align integral_closure.is_integral_closure integralClosure.isIntegralClosureₓ'. -/
 instance integralClosure.isIntegralClosure (R A : Type _) [CommRing R] [CommRing A] [Algebra R A] :
     IsIntegralClosure (integralClosure R A) R A :=
   ⟨Subtype.coe_injective, fun x =>
@@ -850,15 +1274,33 @@ variable [Algebra R B] [Algebra A B] [IsIntegralClosure A R B]
 
 variable (R) {A} (B)
 
+/- warning: is_integral_closure.is_integral -> IsIntegralClosure.isIntegral is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} (B : Type.{u3}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_7))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] (x : A), IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_7 x
+but is expected to have type
+  forall (R : Type.{u3}) {A : Type.{u2}} (B : Type.{u1}) [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u1} B] [_inst_4 : Algebra.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_5 : Algebra.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u3, u1} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u3, u2, u1} R A B (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_7) (Algebra.toSMul.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_5) (Algebra.toSMul.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_4)] (x : A), IsIntegral.{u3, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_7 x
+Case conversion may be inaccurate. Consider using '#align is_integral_closure.is_integral IsIntegralClosure.isIntegralₓ'. -/
 protected theorem isIntegral [Algebra R A] [IsScalarTower R A B] (x : A) : IsIntegral R x :=
   (isIntegral_algebraMap_iff (algebraMap_injective A R B)).mp <|
     show IsIntegral R (algebraMap A B x) from isIntegral_iff.mpr ⟨x, rfl⟩
 #align is_integral_closure.is_integral IsIntegralClosure.isIntegral
 
+/- warning: is_integral_closure.is_integral_algebra -> IsIntegralClosure.isIntegral_algebra is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} (B : Type.{u3}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_7))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))], Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_7
+but is expected to have type
+  forall (R : Type.{u3}) {A : Type.{u2}} (B : Type.{u1}) [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u1} B] [_inst_4 : Algebra.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_5 : Algebra.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u3, u1} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u3, u2, u1} R A B (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_7) (Algebra.toSMul.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_5) (Algebra.toSMul.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_4)], Algebra.IsIntegral.{u3, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_7
+Case conversion may be inaccurate. Consider using '#align is_integral_closure.is_integral_algebra IsIntegralClosure.isIntegral_algebraₓ'. -/
 theorem isIntegral_algebra [Algebra R A] [IsScalarTower R A B] : Algebra.IsIntegral R A := fun x =>
   IsIntegralClosure.isIntegral R B x
 #align is_integral_closure.is_integral_algebra IsIntegralClosure.isIntegral_algebra
 
+/- warning: is_integral_closure.no_zero_smul_divisors -> IsIntegralClosure.noZeroSMulDivisors is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) {A : Type.{u2}} (B : Type.{u3}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_7))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] [_inst_9 : NoZeroSMulDivisors.{u1, u3} R B (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (MulZeroClass.toHasZero.{u3} B (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} B (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} B (NonAssocRing.toNonUnitalNonAssocRing.{u3} B (Ring.toNonAssocRing.{u3} B (CommRing.toRing.{u3} B _inst_3)))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))], NoZeroSMulDivisors.{u1, u2} R A (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (MulZeroClass.toHasZero.{u2} A (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} A (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u2} A (NonAssocRing.toNonUnitalNonAssocRing.{u2} A (Ring.toNonAssocRing.{u2} A (CommRing.toRing.{u2} A _inst_2)))))) (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_7)))))
+but is expected to have type
+  forall (R : Type.{u3}) {A : Type.{u2}} (B : Type.{u1}) [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u1} B] [_inst_4 : Algebra.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_5 : Algebra.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u3, u1} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u3, u2, u1} R A B (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_7) (Algebra.toSMul.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_5) (Algebra.toSMul.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_4)] [_inst_9 : NoZeroSMulDivisors.{u3, u1} R B (CommMonoidWithZero.toZero.{u3} R (CommSemiring.toCommMonoidWithZero.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (CommMonoidWithZero.toZero.{u1} B (CommSemiring.toCommMonoidWithZero.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))) (Algebra.toSMul.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_4)], NoZeroSMulDivisors.{u3, u2} R A (CommMonoidWithZero.toZero.{u3} R (CommSemiring.toCommMonoidWithZero.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (CommMonoidWithZero.toZero.{u2} A (CommSemiring.toCommMonoidWithZero.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_7)
+Case conversion may be inaccurate. Consider using '#align is_integral_closure.no_zero_smul_divisors IsIntegralClosure.noZeroSMulDivisorsₓ'. -/
 theorem noZeroSMulDivisors [Algebra R A] [IsScalarTower R A B] [NoZeroSMulDivisors R B] :
     NoZeroSMulDivisors R A :=
   by
@@ -870,38 +1312,76 @@ theorem noZeroSMulDivisors [Algebra R A] [IsScalarTower R A B] [NoZeroSMulDiviso
 
 variable {R} (A) {B}
 
+#print IsIntegralClosure.mk' /-
 /-- If `x : B` is integral over `R`, then it is an element of the integral closure of `R` in `B`. -/
 noncomputable def mk' (x : B) (hx : IsIntegral R x) : A :=
   Classical.choose (isIntegral_iff.mp hx)
 #align is_integral_closure.mk' IsIntegralClosure.mk'
+-/
 
+/- warning: is_integral_closure.algebra_map_mk' -> IsIntegralClosure.algebraMap_mk' is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} (A : Type.{u2}) {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] (x : B) (hx : IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_4 x), Eq.{succ u3} B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) (IsIntegralClosure.mk'.{u1, u2, u3} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 x hx)) x
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align is_integral_closure.algebra_map_mk' IsIntegralClosure.algebraMap_mk'ₓ'. -/
 @[simp]
 theorem algebraMap_mk' (x : B) (hx : IsIntegral R x) : algebraMap A B (mk' A x hx) = x :=
   Classical.choose_spec (isIntegral_iff.mp hx)
 #align is_integral_closure.algebra_map_mk' IsIntegralClosure.algebraMap_mk'
 
+/- warning: is_integral_closure.mk'_one -> IsIntegralClosure.mk'_one is a dubious translation:
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 @[simp]
 theorem mk'_one (h : IsIntegral R (1 : B) := isIntegral_one) : mk' A 1 h = 1 :=
   algebraMap_injective A R B <| by rw [algebra_map_mk', RingHom.map_one]
 #align is_integral_closure.mk'_one IsIntegralClosure.mk'_one
 
+/- warning: is_integral_closure.mk'_zero -> IsIntegralClosure.mk'_zero is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_integral_closure.mk'_zero IsIntegralClosure.mk'_zeroₓ'. -/
 @[simp]
 theorem mk'_zero (h : IsIntegral R (0 : B) := isIntegral_zero) : mk' A 0 h = 0 :=
   algebraMap_injective A R B <| by rw [algebra_map_mk', RingHom.map_zero]
 #align is_integral_closure.mk'_zero IsIntegralClosure.mk'_zero
 
+/- warning: is_integral_closure.mk'_add -> IsIntegralClosure.mk'_add is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align is_integral_closure.mk'_add IsIntegralClosure.mk'_addₓ'. -/
 @[simp]
 theorem mk'_add (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
     mk' A (x + y) (isIntegral_add hx hy) = mk' A x hx + mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebra_map_mk', RingHom.map_add]
 #align is_integral_closure.mk'_add IsIntegralClosure.mk'_add
 
+/- warning: is_integral_closure.mk'_mul -> IsIntegralClosure.mk'_mul is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u3}} (A : Type.{u1}) {B : Type.{u2}} [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : CommRing.{u2} B] [_inst_4 : Algebra.{u3, u2} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_5 : Algebra.{u1, u2} A B (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_6 : IsIntegralClosure.{u1, u3, u2} A R B _inst_1 (CommRing.toCommSemiring.{u1} A _inst_2) _inst_3 _inst_4 _inst_5] (x : B) (y : B) (hx : IsIntegral.{u3, u2} R B _inst_1 (CommRing.toRing.{u2} B _inst_3) _inst_4 x) (hy : IsIntegral.{u3, u2} R B _inst_1 (CommRing.toRing.{u2} B _inst_3) _inst_4 y), Eq.{succ u1} A (IsIntegralClosure.mk'.{u3, u1, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 (HMul.hMul.{u2, u2, u2} B B B (instHMul.{u2} B (NonUnitalNonAssocRing.toMul.{u2} B (NonAssocRing.toNonUnitalNonAssocRing.{u2} B (Ring.toNonAssocRing.{u2} B (CommRing.toRing.{u2} B _inst_3))))) x y) (isIntegral_mul.{u2, u3} R B _inst_1 _inst_3 _inst_4 x y hx hy)) (HMul.hMul.{u1, u1, u1} A A A (instHMul.{u1} A (NonUnitalNonAssocRing.toMul.{u1} A (NonAssocRing.toNonUnitalNonAssocRing.{u1} A (Ring.toNonAssocRing.{u1} A (CommRing.toRing.{u1} A _inst_2))))) (IsIntegralClosure.mk'.{u3, u1, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 x hx) (IsIntegralClosure.mk'.{u3, u1, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 y hy))
+Case conversion may be inaccurate. Consider using '#align is_integral_closure.mk'_mul IsIntegralClosure.mk'_mulₓ'. -/
 @[simp]
 theorem mk'_mul (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
     mk' A (x * y) (isIntegral_mul hx hy) = mk' A x hx * mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebra_map_mk', RingHom.map_mul]
 #align is_integral_closure.mk'_mul IsIntegralClosure.mk'_mul
 
+/- warning: is_integral_closure.mk'_algebra_map -> IsIntegralClosure.mk'_algebraMap is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u3}} (A : Type.{u2}) {B : Type.{u1}} [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u1} B] [_inst_4 : Algebra.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_5 : Algebra.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u3, u1} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] [_inst_7 : Algebra.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] [_inst_8 : IsScalarTower.{u3, u2, u1} R A B (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_7) (Algebra.toSMul.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) 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_inst_3))))))) (algebraMap.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_4) x)) (isIntegral_algebraMap.{u1, u3} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => B) x) _inst_1 (CommRing.toRing.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => B) x) _inst_3) _inst_4 x)), Eq.{succ u2} A (IsIntegralClosure.mk'.{u3, u2, u1} R A ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => B) x) _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (RingHom.{u3, u1} R B (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => B) _x) (MulHomClass.toFunLike.{max u3 u1, u3, u1} (RingHom.{u3, u1} R B 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+Case conversion may be inaccurate. Consider using '#align is_integral_closure.mk'_algebra_map IsIntegralClosure.mk'_algebraMapₓ'. -/
 @[simp]
 theorem mk'_algebraMap [Algebra R A] [IsScalarTower R A B] (x : R)
     (h : IsIntegral R (algebraMap R B x) := isIntegral_algebraMap) :
@@ -915,6 +1395,7 @@ variable {R} (A B) {S : Type _} [CommRing S] [Algebra R S] [Algebra S B] [IsScal
 
 variable [Algebra R A] [IsScalarTower R A B] (h : Algebra.IsIntegral R S)
 
+#print IsIntegralClosure.lift /-
 /-- If `B / S / R` is a tower of ring extensions where `S` is integral over `R`,
 then `S` maps (uniquely) into an integral closure `B / A / R`. -/
 noncomputable def lift : S →ₐ[R] A
@@ -926,7 +1407,14 @@ noncomputable def lift : S →ₐ[R] A
   map_mul' x y := by simp_rw [← mk'_mul, RingHom.map_mul]
   commutes' x := by simp_rw [← IsScalarTower.algebraMap_apply, mk'_algebra_map]
 #align is_integral_closure.lift IsIntegralClosure.lift
+-/
 
+/- warning: is_integral_closure.algebra_map_lift -> IsIntegralClosure.algebraMap_lift is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} (A : Type.{u2}) (B : Type.{u3}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] {S : Type.{u4}} [_inst_7 : CommRing.{u4} S] [_inst_8 : Algebra.{u1, u4} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7))] [_inst_9 : Algebra.{u4, u3} S B (CommRing.toCommSemiring.{u4} S _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_10 : IsScalarTower.{u1, u4, u3} R S B (SMulZeroClass.toHasSmul.{u1, u4} R S (AddZeroClass.toHasZero.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)))))))) (SMulWithZero.toSmulZeroClass.{u1, u4} 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.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)))))))) (MulActionWithZero.toSMulWithZero.{u1, u4} R S (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u4} S (AddMonoid.toAddZeroClass.{u4} S (AddCommMonoid.toAddMonoid.{u4} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)))))))) (Module.toMulActionWithZero.{u1, u4} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7))))) (Algebra.toModule.{u1, u4} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) _inst_8))))) (SMulZeroClass.toHasSmul.{u4, u3} S B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u4, u3} S B (MulZeroClass.toHasZero.{u4} S (MulZeroOneClass.toMulZeroClass.{u4} S (MonoidWithZero.toMulZeroOneClass.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u4, u3} S B (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u4, u3} S B (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u4, u3} S B (CommRing.toCommSemiring.{u4} S _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_9))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] [_inst_11 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_12 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_11))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] (h : Algebra.IsIntegral.{u1, u4} R S _inst_1 (CommRing.toRing.{u4} S _inst_7) _inst_8) (x : S), Eq.{succ u3} B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) (coeFn.{max (succ u4) (succ u2), max (succ u4) (succ u2)} (AlgHom.{u1, u4, u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8 _inst_11) (fun (_x : AlgHom.{u1, u4, u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8 _inst_11) => S -> A) ([anonymous].{u1, u4, u2} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} S (CommRing.toRing.{u4} S _inst_7)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8 _inst_11) (IsIntegralClosure.lift.{u1, u2, u3, u4} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 S _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 h) x)) (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (RingHom.{u4, u3} S B (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u4, u3} S B (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => S -> B) (RingHom.hasCoeToFun.{u4, u3} S B (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S (CommRing.toCommSemiring.{u4} S _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u4, u3} S B (CommRing.toCommSemiring.{u4} S _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_9) x)
+but is expected to have type
+  forall {R : Type.{u1}} (A : Type.{u3}) (B : Type.{u4}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u4} B] [_inst_4 : Algebra.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_5 : Algebra.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_6 : IsIntegralClosure.{u3, u1, u4} A R B _inst_1 (CommRing.toCommSemiring.{u3} A _inst_2) _inst_3 _inst_4 _inst_5] {S : Type.{u2}} [_inst_7 : CommRing.{u2} S] [_inst_8 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))] [_inst_9 : Algebra.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_10 : IsScalarTower.{u1, u2, u4} R S B (Algebra.toSMul.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8) (Algebra.toSMul.{u2, u4} S B (CommRing.toCommSemiring.{u2} S _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_9) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] [_inst_11 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_12 : IsScalarTower.{u1, u3, u4} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11) (Algebra.toSMul.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] (h : Algebra.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_7) _inst_8) (x : S), Eq.{succ u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) S (fun (a : S) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : S) => A) a) (SMulHomClass.toFunLike.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (SMulZeroClass.toSMul.{u1, u2} R S (AddMonoid.toZero.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R S (AddMonoid.toAddZeroClass.{u2} S (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R S (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8))))) (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} S (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)))))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8)) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u2, u3} (AlgHom.{u1, u2, u3} R S A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8 _inst_11) R S A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{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_7)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (Module.toDistribMulAction.{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 (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7))))) (Algebra.toModule.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_7)) _inst_8)) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_11)) 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+Case conversion may be inaccurate. Consider using '#align is_integral_closure.algebra_map_lift IsIntegralClosure.algebraMap_liftₓ'. -/
 @[simp]
 theorem algebraMap_lift (x : S) : algebraMap A B (lift A B h x) = algebraMap S B x :=
   algebraMap_mk' _ _ _
@@ -940,6 +1428,7 @@ variable (R A B) (A' : Type _) [CommRing A'] [Algebra A' B] [IsIntegralClosure A
 
 variable [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A' B]
 
+#print IsIntegralClosure.equiv /-
 /-- Integral closures are all isomorphic to each other. -/
 noncomputable def equiv : A ≃ₐ[R] A' :=
   AlgEquiv.ofAlgHom (lift _ B (isIntegral_algebra R B)) (lift _ B (isIntegral_algebra R B))
@@ -952,7 +1441,14 @@ noncomputable def equiv : A ≃ₐ[R] A' :=
       apply algebra_map_injective A R B
       simp)
 #align is_integral_closure.equiv IsIntegralClosure.equiv
+-/
 
+/- warning: is_integral_closure.algebra_map_equiv -> IsIntegralClosure.algebraMap_equiv is a dubious translation:
+lean 3 declaration is
+  forall (R : Type.{u1}) (A : Type.{u2}) (B : Type.{u3}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_4 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_5 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_6 : IsIntegralClosure.{u2, u1, u3} A R B _inst_1 (CommRing.toCommSemiring.{u2} A _inst_2) _inst_3 _inst_4 _inst_5] (A' : Type.{u4}) [_inst_7 : CommRing.{u4} A'] [_inst_8 : Algebra.{u4, u3} A' B (CommRing.toCommSemiring.{u4} A' _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_9 : IsIntegralClosure.{u4, u1, u3} A' R B _inst_1 (CommRing.toCommSemiring.{u4} A' _inst_7) _inst_3 _inst_4 _inst_8] [_inst_10 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_11 : Algebra.{u1, u4} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7))] [_inst_12 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_10))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] [_inst_13 : IsScalarTower.{u1, u4, u3} R A' B (SMulZeroClass.toHasSmul.{u1, u4} R A' (AddZeroClass.toHasZero.{u4} A' (AddMonoid.toAddZeroClass.{u4} A' (AddCommMonoid.toAddMonoid.{u4} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)))))))) (SMulWithZero.toSmulZeroClass.{u1, u4} R A' (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.{u4} A' (AddMonoid.toAddZeroClass.{u4} A' (AddCommMonoid.toAddMonoid.{u4} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)))))))) (MulActionWithZero.toSMulWithZero.{u1, u4} R A' (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u4} A' (AddMonoid.toAddZeroClass.{u4} A' (AddCommMonoid.toAddMonoid.{u4} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)))))))) (Module.toMulActionWithZero.{u1, u4} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} A' (Semiring.toNonAssocSemiring.{u4} A' (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7))))) (Algebra.toModule.{u1, u4} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_11))))) (SMulZeroClass.toHasSmul.{u4, u3} A' B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u4, u3} A' B (MulZeroClass.toHasZero.{u4} A' (MulZeroOneClass.toMulZeroClass.{u4} A' (MonoidWithZero.toMulZeroOneClass.{u4} A' (Semiring.toMonoidWithZero.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u4, u3} A' B (Semiring.toMonoidWithZero.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u4, u3} A' B (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u4, u3} A' B (CommRing.toCommSemiring.{u4} A' _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_8))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_4)))))] (x : A), Eq.{succ u3} B (coeFn.{max (succ u4) (succ u3), max (succ u4) (succ u3)} (RingHom.{u4, u3} A' B (Semiring.toNonAssocSemiring.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u4, u3} A' B (Semiring.toNonAssocSemiring.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A' -> B) (RingHom.hasCoeToFun.{u4, u3} A' B (Semiring.toNonAssocSemiring.{u4} A' (CommSemiring.toSemiring.{u4} A' (CommRing.toCommSemiring.{u4} A' _inst_7))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u4, u3} A' B (CommRing.toCommSemiring.{u4} A' _inst_7) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_8) (coeFn.{max (succ u2) (succ u4), max (succ u2) (succ u4)} (AlgEquiv.{u1, u2, u4} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_10 _inst_11) (fun (_x : AlgEquiv.{u1, u2, u4} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_10 _inst_11) => A -> A') (AlgEquiv.hasCoeToFun.{u1, u2, u4} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (Ring.toSemiring.{u4} A' (CommRing.toRing.{u4} A' _inst_7)) _inst_10 _inst_11) (IsIntegralClosure.equiv.{u1, u2, u3, u4} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 A' _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13) x)) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_5) x)
+but is expected to have type
+  forall (R : Type.{u1}) (A : Type.{u3}) (B : Type.{u4}) [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u4} B] [_inst_4 : Algebra.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_5 : Algebra.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_6 : IsIntegralClosure.{u3, u1, u4} A R B _inst_1 (CommRing.toCommSemiring.{u3} A _inst_2) _inst_3 _inst_4 _inst_5] (A' : Type.{u2}) [_inst_7 : CommRing.{u2} A'] [_inst_8 : Algebra.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))] [_inst_9 : IsIntegralClosure.{u2, u1, u4} A' R B _inst_1 (CommRing.toCommSemiring.{u2} A' _inst_7) _inst_3 _inst_4 _inst_8] [_inst_10 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_11 : Algebra.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))] [_inst_12 : IsScalarTower.{u1, u3, u4} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10) (Algebra.toSMul.{u3, u4} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_5) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] [_inst_13 : IsScalarTower.{u1, u2, u4} R A' B (Algebra.toSMul.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11) (Algebra.toSMul.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_8) (Algebra.toSMul.{u1, u4} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_4)] (x : A), Eq.{succ u4} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A') => B) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) A (fun (a : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => A') a) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10))))) (SMulZeroClass.toSMul.{u1, u2} R A' (AddMonoid.toZero.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribSMul.toSMulZeroClass.{u1, u2} R A' (AddMonoid.toAddZeroClass.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))))) (DistribMulAction.toDistribSMul.{u1, u2} R A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11))))) (DistribMulActionHomClass.toSMulHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10)) (Module.toDistribMulAction.{u1, u2} R A' (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (Algebra.toModule.{u1, u2} R A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_11)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u1, u3, u2, max u3 u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) (AlgEquivClass.toAlgHomClass.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11 (AlgEquiv.instAlgEquivClassAlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11)))))) (IsIntegralClosure.equiv.{u1, u3, u4, u2} R A B _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 A' _inst_7 _inst_8 _inst_9 _inst_10 _inst_11 _inst_12 _inst_13) x)) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' (fun (_x : A') => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A') => B) _x) (MulHomClass.toFunLike.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (NonUnitalNonAssocSemiring.toMul.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))))) (NonUnitalNonAssocSemiring.toMul.{u4} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} B (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u4 u2, u2, u4} (RingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)))) A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))) (RingHom.instRingHomClassRingHom.{u2, u4} A' B (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7))) (Semiring.toNonAssocSemiring.{u4} B (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3))))))) (algebraMap.{u2, u4} A' B (CommRing.toCommSemiring.{u2} A' _inst_7) (CommSemiring.toSemiring.{u4} B (CommRing.toCommSemiring.{u4} B _inst_3)) _inst_8) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : A) => A') _x) (SMulHomClass.toFunLike.{max u3 u2, u1, u3, u2} (AlgEquiv.{u1, u3, u2} R A A' (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} A' (CommRing.toCommSemiring.{u2} A' _inst_7)) _inst_10 _inst_11) R A A' (SMulZeroClass.toSMul.{u1, u3} R A (AddMonoid.toZero.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribSMul.toSMulZeroClass.{u1, u3} R A (AddMonoid.toAddZeroClass.{u3} A (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (DistribMulAction.toDistribSMul.{u1, u3} R A (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)))) (AddCommMonoid.toAddMonoid.{u3} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Module.toDistribMulAction.{u1, u3} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Algebra.toModule.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_10))))) (SMulZeroClass.toSMul.{u1, u2} R A' (AddMonoid.toZero.{u2} A' (AddCommMonoid.toAddMonoid.{u2} A' (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A' (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A' (Semiring.toNonAssocSemiring.{u2} A' (CommSemiring.toSemiring.{u2} A' 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+Case conversion may be inaccurate. Consider using '#align is_integral_closure.algebra_map_equiv IsIntegralClosure.algebraMap_equivₓ'. -/
 @[simp]
 theorem algebraMap_equiv (x : A) : algebraMap A' B (equiv R A B A' x) = algebraMap A B x :=
   algebraMap_lift _ _ _ _
@@ -974,6 +1470,12 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S] [CommRing T]
 
 variable [Algebra A B] [Algebra R B] (f : R →+* S) (g : S →+* T)
 
+/- warning: is_integral_trans_aux -> isIntegral_trans_aux is a dubious translation:
+lean 3 declaration is
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(CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7) Type.{u3} (SetLike.hasCoeToSort.{u3, u3} (Subalgebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7) B (Subalgebra.setLike.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7)) (Algebra.adjoin.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7 ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} B) (Set.{u3} B) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (Finset.Set.hasCoeT.{u3} B))) (Polynomial.frange.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (Polynomial.map.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6) p))))) B (Subalgebra.toCommRing.{u1, u3} R B _inst_1 _inst_3 _inst_7 (Algebra.adjoin.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7 ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} B) (Set.{u3} B) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (Finset.Set.hasCoeT.{u3} B))) (Polynomial.frange.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (Polynomial.map.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6) p))))) (CommRing.toRing.{u3} B _inst_3) (Subalgebra.toAlgebra.{u3, u1, u3} B R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u3} B _inst_3) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7 (Algebra.id.{u3} B (CommRing.toCommSemiring.{u3} B _inst_3)) (Algebra.adjoin.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7 ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Finset.{u3} B) (Set.{u3} B) (HasLiftT.mk.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (CoeTCₓ.coe.{succ u3, succ u3} (Finset.{u3} B) (Set.{u3} B) (Finset.Set.hasCoeT.{u3} B))) (Polynomial.frange.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (Polynomial.map.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6) p))))) x)
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u3}} {B : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u2} B] [_inst_6 : Algebra.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_7 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] (x : B) {p : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))}, (Polynomial.Monic.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) p) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (FunLike.coe.{max (succ u2) (succ u3), succ u3, succ u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (fun (_x : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) _x) (SMulHomClass.toFunLike.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (SMulZeroClass.toSMul.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddMonoid.toZero.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (DistribSMul.toSMulZeroClass.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddMonoid.toAddZeroClass.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (DistribMulAction.toDistribSMul.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))))) (SMulZeroClass.toSMul.{u3, u2} A B (AddMonoid.toZero.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (DistribSMul.toSMulZeroClass.{u3, u2} A B (AddMonoid.toAddZeroClass.{u2} B (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (DistribMulAction.toDistribSMul.{u3, u2} A B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6))))) (DistribMulActionHomClass.toSMulHomClass.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (AddCommMonoid.toAddMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))))) (AddCommMonoid.toAddMonoid.{u2} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)) (NonUnitalAlgHomClass.toDistribMulActionHomClass.{max u2 u3, u3, u3, u2} (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (MonoidWithZero.toMonoid.{u3} A (Semiring.toMonoidWithZero.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (Module.toDistribMulAction.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))))) (Algebra.toModule.{u3, u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (Module.toDistribMulAction.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (Algebra.toModule.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6)) (AlgHom.instNonUnitalAlgHomClassToMonoidToMonoidWithZeroToSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToNonUnitalNonAssocSemiringToNonAssocSemiringToDistribMulActionToAddCommMonoidToModuleToDistribMulActionToAddCommMonoidToModule.{u3, u3, u2, max u2 u3} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6 (AlgHom.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6) (AlgHom.algHomClass.{u3, u3, u2} A (Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) B (CommRing.toCommSemiring.{u3} A _inst_2) (Polynomial.semiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.algebraOfAlgebra.{u3, u3} A A (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (Algebra.id.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) _inst_6))))) (Polynomial.aeval.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6 x) p) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommMonoidWithZero.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommSemiring.toCommMonoidWithZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) (CommRing.toCommSemiring.{u2} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Polynomial.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) => B) p) _inst_3)))))) -> (IsIntegral.{u2, u2} (Subtype.{succ u2} B (fun (x : B) => Membership.mem.{u2, u2} B (Subalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) (SetLike.instMembership.{u2, u2} (Subalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7) B (Subalgebra.instSetLikeSubalgebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7)) x (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p)))))) B (Subalgebra.toCommRing.{u1, u2} R B _inst_1 _inst_3 _inst_7 (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p))))) (CommRing.toRing.{u2} B _inst_3) (Subalgebra.toAlgebra.{u2, u1, u2} B R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommRing.toCommSemiring.{u2} B _inst_3) (Ring.toSemiring.{u2} B (CommRing.toRing.{u2} B _inst_3)) _inst_7 (Algebra.id.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Algebra.adjoin.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7 (Finset.toSet.{u2} B (Polynomial.frange.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (Polynomial.map.{u3, u2} A B (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) p))))) x)
+Case conversion may be inaccurate. Consider using '#align is_integral_trans_aux isIntegral_trans_auxₓ'. -/
 theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x p = 0) :
     IsIntegral (adjoin R (↑(p.map <| algebraMap A B).frange : Set B)) x :=
   by
@@ -1005,22 +1507,34 @@ theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x
 
 variable [Algebra R A] [IsScalarTower R A B]
 
+/- warning: is_integral_trans -> isIntegral_trans is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_6 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_7 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_8 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_9 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7)))))], (Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8) -> (forall (x : B), (IsIntegral.{u2, u3} A B _inst_2 (CommRing.toRing.{u3} B _inst_3) _inst_6 x) -> (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_7 x))
+but is expected to have type
+  forall {R : Type.{u3}} {A : Type.{u2}} {B : Type.{u1}} [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u1} B] [_inst_6 : Algebra.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_7 : Algebra.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_8 : Algebra.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] [_inst_9 : IsScalarTower.{u3, u2, u1} R A B (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_8) (Algebra.toSMul.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_6) (Algebra.toSMul.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_7)], (Algebra.IsIntegral.{u3, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8) -> (forall (x : B), (IsIntegral.{u2, u1} A B _inst_2 (CommRing.toRing.{u1} B _inst_3) _inst_6 x) -> (IsIntegral.{u3, u1} R B _inst_1 (CommRing.toRing.{u1} B _inst_3) _inst_7 x))
+Case conversion may be inaccurate. Consider using '#align is_integral_trans isIntegral_transₓ'. -/
 /-- If A is an R-algebra all of whose elements are integral over R,
 and x is an element of an A-algebra that is integral over A, then x is integral over R.-/
 theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegral A x) :
     IsIntegral R x := by
   rcases hx with ⟨p, pmonic, hp⟩
   let S : Set B := ↑(p.map <| algebraMap A B).frange
-  refine' isIntegral_of_mem_of_fG (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
-  refine' fg_trans (fG_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
+  refine' isIntegral_of_mem_of_FG (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
+  refine' fg_trans (FG_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
   · rw [Finset.mem_coe, frange, Finset.mem_image] at hx
     rcases hx with ⟨i, _, rfl⟩
     rw [coeff_map]
     exact map_isIntegral (IsScalarTower.toAlgHom R A B) (A_int _)
-  · apply fG_adjoin_singleton_of_integral
+  · apply FG_adjoin_singleton_of_integral
     exact isIntegral_trans_aux _ pmonic hp
 #align is_integral_trans isIntegral_trans
 
+/- warning: algebra.is_integral_trans -> Algebra.isIntegral_trans is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_6 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_7 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_8 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_9 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7)))))], (Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8) -> (Algebra.IsIntegral.{u2, u3} A B _inst_2 (CommRing.toRing.{u3} B _inst_3) _inst_6) -> (Algebra.IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_7)
+but is expected to have type
+  forall {R : Type.{u3}} {A : Type.{u2}} {B : Type.{u1}} [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u1} B] [_inst_6 : Algebra.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_7 : Algebra.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3))] [_inst_8 : Algebra.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] [_inst_9 : IsScalarTower.{u3, u2, u1} R A B (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_8) (Algebra.toSMul.{u2, u1} A B (CommRing.toCommSemiring.{u2} A _inst_2) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_6) (Algebra.toSMul.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_3)) _inst_7)], (Algebra.IsIntegral.{u3, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8) -> (Algebra.IsIntegral.{u2, u1} A B _inst_2 (CommRing.toRing.{u1} B _inst_3) _inst_6) -> (Algebra.IsIntegral.{u3, u1} R B _inst_1 (CommRing.toRing.{u1} B _inst_3) _inst_7)
+Case conversion may be inaccurate. Consider using '#align algebra.is_integral_trans Algebra.isIntegral_transₓ'. -/
 /-- If A is an R-algebra all of whose elements are integral over R,
 and B is an A-algebra all of whose elements are integral over A,
 then all elements of B are integral over R.-/
@@ -1028,6 +1542,12 @@ theorem Algebra.isIntegral_trans (hA : Algebra.IsIntegral R A) (hB : Algebra.IsI
     Algebra.IsIntegral R B := fun x => isIntegral_trans hA x (hB x)
 #align algebra.is_integral_trans Algebra.isIntegral_trans
 
+/- warning: ring_hom.is_integral_trans -> RingHom.isIntegral_trans is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} {T : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] [_inst_5 : CommRing.{u3} T] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) (g : RingHom.{u2, u3} S T (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4))) (NonAssocRing.toNonAssocSemiring.{u3} T (Ring.toNonAssocRing.{u3} T (CommRing.toRing.{u3} T _inst_5)))), (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f) -> (RingHom.IsIntegral.{u2, u3} S T _inst_4 (CommRing.toRing.{u3} T _inst_5) g) -> (RingHom.IsIntegral.{u1, u3} R T _inst_1 (CommRing.toRing.{u3} T _inst_5) (RingHom.comp.{u1, u2, u3} R S T (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4))) (NonAssocRing.toNonAssocSemiring.{u3} T (Ring.toNonAssocRing.{u3} T (CommRing.toRing.{u3} T _inst_5))) g f))
+but is expected to have type
+  forall {R : Type.{u3}} {S : Type.{u2}} {T : Type.{u1}} [_inst_1 : CommRing.{u3} R] [_inst_4 : CommRing.{u2} S] [_inst_5 : CommRing.{u1} T] (f : RingHom.{u3, u2} R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)))) (g : RingHom.{u2, u1} S T (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))) (Semiring.toNonAssocSemiring.{u1} T (CommSemiring.toSemiring.{u1} T (CommRing.toCommSemiring.{u1} T _inst_5)))), (RingHom.IsIntegral.{u3, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f) -> (RingHom.IsIntegral.{u2, u1} S T _inst_4 (CommRing.toRing.{u1} T _inst_5) g) -> (RingHom.IsIntegral.{u3, u1} R T _inst_1 (CommRing.toRing.{u1} T _inst_5) (RingHom.comp.{u3, u2, u1} R S T (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))) (Semiring.toNonAssocSemiring.{u1} T (CommSemiring.toSemiring.{u1} T (CommRing.toCommSemiring.{u1} T _inst_5))) g f))
+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_trans RingHom.isIntegral_transₓ'. -/
 theorem RingHom.isIntegral_trans (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.comp f).IsIntegral :=
   @Algebra.isIntegral_trans R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
     (@IsScalarTower.of_algebraMap_eq R S T _ _ _ f.toAlgebra g.toAlgebra (g.comp f).toAlgebra
@@ -1035,15 +1555,33 @@ theorem RingHom.isIntegral_trans (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.co
     hf hg
 #align ring_hom.is_integral_trans RingHom.isIntegral_trans
 
+/- warning: ring_hom.is_integral_of_surjective -> RingHom.isIntegral_of_surjective is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))), (Function.Surjective.{succ u1, succ u2} R S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) (fun (_x : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) f)) -> (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f)
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_of_surjective RingHom.isIntegral_of_surjectiveₓ'. -/
 theorem RingHom.isIntegral_of_surjective (hf : Function.Surjective f) : f.IsIntegral := fun x =>
   (hf x).recOn fun y hy => (hy ▸ f.is_integral_map : f.IsIntegralElem x)
 #align ring_hom.is_integral_of_surjective RingHom.isIntegral_of_surjective
 
+/- warning: is_integral_of_surjective -> isIntegral_of_surjective is a dubious translation:
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+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_8 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))], (Function.Surjective.{succ u1, succ u2} R A (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))) (fun (_x : RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))) => R -> A) (RingHom.hasCoeToFun.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))) (algebraMap.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8))) -> (Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8)
+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_8 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))], (Function.Surjective.{succ u2, succ u1} R A (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} R A (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : R) => A) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} R A (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) R A (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} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R A (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) R A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} A (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} R A (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)))) R A (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))) (RingHom.instRingHomClassRingHom.{u2, u1} R A (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} A (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))))))) (algebraMap.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_8))) -> (Algebra.IsIntegral.{u2, u1} R A _inst_1 (CommRing.toRing.{u1} A _inst_2) _inst_8)
+Case conversion may be inaccurate. Consider using '#align is_integral_of_surjective isIntegral_of_surjectiveₓ'. -/
 theorem isIntegral_of_surjective (h : Function.Surjective (algebraMap R A)) :
     Algebra.IsIntegral R A :=
   (algebraMap R A).isIntegral_of_surjective h
 #align is_integral_of_surjective isIntegral_of_surjective
 
+/- warning: is_integral_tower_bot_of_is_integral -> isIntegral_tower_bot_of_isIntegral is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_6 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_7 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_8 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_9 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7)))))], (Function.Injective.{succ u2, succ u3} A B (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6))) -> (forall {x : A}, (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_7 (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))) (algebraMap.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6) x)) -> (IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8 x))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u3}} {B : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u3} A] [_inst_3 : CommRing.{u2} B] [_inst_6 : Algebra.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_7 : Algebra.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_8 : Algebra.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))] [_inst_9 : IsScalarTower.{u1, u3, u2} R A B (Algebra.toSMul.{u1, u3} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)) _inst_8) (Algebra.toSMul.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) (Algebra.toSMul.{u1, u2} R B (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7)], (Function.Injective.{succ u3, succ u2} A B (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonUnitalNonAssocSemiring.toMul.{u2} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))) (NonUnitalRingHomClass.toMulHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} B (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) (RingHomClass.toNonUnitalRingHomClass.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6))) -> (forall {x : A}, (IsIntegral.{u1, u2} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_1 (CommRing.toRing.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_3) _inst_7 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u3 u2, u3, u2} (RingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (NonUnitalNonAssocSemiring.toMul.{u3} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} A (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))))) (NonUnitalNonAssocSemiring.toMul.{u2} B 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(Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)))) A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))) (RingHom.instRingHomClassRingHom.{u3, u2} A B (Semiring.toNonAssocSemiring.{u3} A (CommSemiring.toSemiring.{u3} A (CommRing.toCommSemiring.{u3} A _inst_2))) (Semiring.toNonAssocSemiring.{u2} B (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))))))) (algebraMap.{u3, u2} A B (CommRing.toCommSemiring.{u3} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) x)) -> (IsIntegral.{u1, u3} R A _inst_1 (CommRing.toRing.{u3} A _inst_2) _inst_8 x))
+Case conversion may be inaccurate. Consider using '#align is_integral_tower_bot_of_is_integral isIntegral_tower_bot_of_isIntegralₓ'. -/
 /-- If `R → A → B` is an algebra tower with `A → B` injective,
 then if the entire tower is an integral extension so is `R → A` -/
 theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A B)) {x : A}
@@ -1057,6 +1595,12 @@ theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A
   exact H hp'
 #align is_integral_tower_bot_of_is_integral isIntegral_tower_bot_of_isIntegral
 
+/- warning: ring_hom.is_integral_tower_bot_of_is_integral -> RingHom.isIntegral_tower_bot_of_isIntegral is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_tower_bot_of_is_integral RingHom.isIntegral_tower_bot_of_isIntegralₓ'. -/
 theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injective g)
     (hfg : (g.comp f).IsIntegral) : f.IsIntegral := fun x =>
   @isIntegral_tower_bot_of_isIntegral R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
@@ -1065,22 +1609,46 @@ theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injective g)
     hg x (hfg (g x))
 #align ring_hom.is_integral_tower_bot_of_is_integral RingHom.isIntegral_tower_bot_of_isIntegral
 
+/- warning: is_integral_tower_bot_of_is_integral_field -> isIntegral_tower_bot_of_isIntegral_field is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_10 : CommRing.{u1} R] [_inst_11 : Field.{u2} A] [_inst_12 : CommRing.{u3} B] [_inst_13 : Nontrivial.{u3} B] [_inst_14 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_10) (Ring.toSemiring.{u2} A (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11)))] [_inst_15 : Algebra.{u2, u3} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12))] [_inst_16 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_10) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12))] [_inst_17 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A 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(NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12))))) (Algebra.toModule.{u2, u3} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)) _inst_15))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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_10)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_10))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_10)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_10) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)) _inst_16)))))] {x : A}, (IsIntegral.{u1, u3} R B _inst_10 (CommRing.toRing.{u3} B _inst_12) _inst_16 (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))) (fun (_x : RingHom.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))) => A -> B) (RingHom.hasCoeToFun.{u2, u3} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)))) (algebraMap.{u2, u3} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_12)) _inst_15) x)) -> (IsIntegral.{u1, u2} R A _inst_10 (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11)) _inst_14 x)
+but is expected to have type
+  forall {R : Type.{u3}} {A : Type.{u2}} {B : Type.{u1}} [_inst_10 : CommRing.{u3} R] [_inst_11 : Field.{u2} A] [_inst_12 : CommRing.{u1} B] [_inst_13 : Nontrivial.{u1} B] [_inst_14 : Algebra.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_10) (DivisionSemiring.toSemiring.{u2} A (Semifield.toDivisionSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))] [_inst_15 : Algebra.{u2, u1} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12))] [_inst_16 : Algebra.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_10) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12))] [_inst_17 : IsScalarTower.{u3, u2, u1} R A B (Algebra.toSMul.{u3, u2} R A (CommRing.toCommSemiring.{u3} R _inst_10) (DivisionSemiring.toSemiring.{u2} A (Semifield.toDivisionSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11))) _inst_14) (Algebra.toSMul.{u2, u1} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)) _inst_15) (Algebra.toSMul.{u3, u1} R B (CommRing.toCommSemiring.{u3} R _inst_10) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)) _inst_16)] {x : A}, (IsIntegral.{u3, u1} R ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_10 (CommRing.toRing.{u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) x) _inst_12) _inst_16 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)))) A (fun (_x : A) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2397 : A) => B) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)))) A B (NonUnitalNonAssocSemiring.toMul.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))))) (NonUnitalNonAssocSemiring.toMul.{u1} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} B (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12))))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)))) A B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11))))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} B (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)))) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)))) A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12))) (RingHom.instRingHomClassRingHom.{u2, u1} A B (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)))) (Semiring.toNonAssocSemiring.{u1} B (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12))))))) (algebraMap.{u2, u1} A B (Semifield.toCommSemiring.{u2} A (Field.toSemifield.{u2} A _inst_11)) (CommSemiring.toSemiring.{u1} B (CommRing.toCommSemiring.{u1} B _inst_12)) _inst_15) x)) -> (IsIntegral.{u3, u2} R A _inst_10 (DivisionRing.toRing.{u2} A (Field.toDivisionRing.{u2} A _inst_11)) _inst_14 x)
+Case conversion may be inaccurate. Consider using '#align is_integral_tower_bot_of_is_integral_field isIntegral_tower_bot_of_isIntegral_fieldₓ'. -/
 theorem isIntegral_tower_bot_of_isIntegral_field {R A B : Type _} [CommRing R] [Field A]
     [CommRing B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]
     {x : A} (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
   isIntegral_tower_bot_of_isIntegral (algebraMap A B).Injective h
 #align is_integral_tower_bot_of_is_integral_field isIntegral_tower_bot_of_isIntegral_field
 
+/- warning: ring_hom.is_integral_elem_of_is_integral_elem_comp -> RingHom.isIntegralElem_of_isIntegralElem_comp is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} {T : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] [_inst_5 : CommRing.{u3} T] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) (g : RingHom.{u2, u3} S T (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4))) (NonAssocRing.toNonAssocSemiring.{u3} T (Ring.toNonAssocRing.{u3} T (CommRing.toRing.{u3} T _inst_5)))) {x : T}, (RingHom.IsIntegralElem.{u1, u3} R T _inst_1 (CommRing.toRing.{u3} T _inst_5) (RingHom.comp.{u1, u2, u3} R S T (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4))) (NonAssocRing.toNonAssocSemiring.{u3} T (Ring.toNonAssocRing.{u3} T (CommRing.toRing.{u3} T _inst_5))) g f) x) -> (RingHom.IsIntegralElem.{u2, u3} S T _inst_4 (CommRing.toRing.{u3} T _inst_5) g x)
+but is expected to have type
+  forall {R : Type.{u3}} {S : Type.{u1}} {T : Type.{u2}} [_inst_1 : CommRing.{u3} R] [_inst_4 : CommRing.{u1} S] [_inst_5 : CommRing.{u2} T] (f : RingHom.{u3, u1} R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)))) (g : RingHom.{u1, u2} S T (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))) (Semiring.toNonAssocSemiring.{u2} T (CommSemiring.toSemiring.{u2} T (CommRing.toCommSemiring.{u2} T _inst_5)))) {x : T}, (RingHom.IsIntegralElem.{u3, u2} R T _inst_1 (CommRing.toRing.{u2} T _inst_5) (RingHom.comp.{u3, u1, u2} R S T (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))) (Semiring.toNonAssocSemiring.{u2} T (CommSemiring.toSemiring.{u2} T (CommRing.toCommSemiring.{u2} T _inst_5))) g f) x) -> (RingHom.IsIntegralElem.{u1, u2} S T _inst_4 (CommRing.toRing.{u2} T _inst_5) g x)
+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem_of_isIntegralElem_compₓ'. -/
 theorem RingHom.isIntegralElem_of_isIntegralElem_comp {x : T} (h : (g.comp f).IsIntegralElem x) :
     g.IsIntegralElem x :=
   let ⟨p, ⟨hp, hp'⟩⟩ := h
   ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp'⟩
 #align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem_of_isIntegralElem_comp
 
+/- warning: ring_hom.is_integral_tower_top_of_is_integral -> RingHom.isIntegral_tower_top_of_isIntegral is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} {T : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] [_inst_5 : CommRing.{u3} T] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) (g : RingHom.{u2, u3} S T (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4))) (NonAssocRing.toNonAssocSemiring.{u3} T (Ring.toNonAssocRing.{u3} T (CommRing.toRing.{u3} T _inst_5)))), (RingHom.IsIntegral.{u1, u3} R T _inst_1 (CommRing.toRing.{u3} T _inst_5) (RingHom.comp.{u1, u2, u3} R S T (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4))) (NonAssocRing.toNonAssocSemiring.{u3} T (Ring.toNonAssocRing.{u3} T (CommRing.toRing.{u3} T _inst_5))) g f)) -> (RingHom.IsIntegral.{u2, u3} S T _inst_4 (CommRing.toRing.{u3} T _inst_5) g)
+but is expected to have type
+  forall {R : Type.{u3}} {S : Type.{u1}} {T : Type.{u2}} [_inst_1 : CommRing.{u3} R] [_inst_4 : CommRing.{u1} S] [_inst_5 : CommRing.{u2} T] (f : RingHom.{u3, u1} R S (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4)))) (g : RingHom.{u1, u2} S T (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))) (Semiring.toNonAssocSemiring.{u2} T (CommSemiring.toSemiring.{u2} T (CommRing.toCommSemiring.{u2} T _inst_5)))), (RingHom.IsIntegral.{u3, u2} R T _inst_1 (CommRing.toRing.{u2} T _inst_5) (RingHom.comp.{u3, u1, u2} R S T (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R (CommRing.toCommSemiring.{u3} R _inst_1))) (Semiring.toNonAssocSemiring.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_4))) (Semiring.toNonAssocSemiring.{u2} T (CommSemiring.toSemiring.{u2} T (CommRing.toCommSemiring.{u2} T _inst_5))) g f)) -> (RingHom.IsIntegral.{u1, u2} S T _inst_4 (CommRing.toRing.{u2} T _inst_5) g)
+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_tower_top_of_is_integral RingHom.isIntegral_tower_top_of_isIntegralₓ'. -/
 theorem RingHom.isIntegral_tower_top_of_isIntegral (h : (g.comp f).IsIntegral) : g.IsIntegral :=
   fun x => RingHom.isIntegralElem_of_isIntegralElem_comp f g (h x)
 #align ring_hom.is_integral_tower_top_of_is_integral RingHom.isIntegral_tower_top_of_isIntegral
 
+/- warning: is_integral_tower_top_of_is_integral -> isIntegral_tower_top_of_isIntegral is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} {B : Type.{u3}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_3 : CommRing.{u3} B] [_inst_6 : Algebra.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_7 : Algebra.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))] [_inst_8 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] [_inst_9 : IsScalarTower.{u1, u2, u3} R A B (SMulZeroClass.toHasSmul.{u1, u2} R A (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (SMulWithZero.toSmulZeroClass.{u1, u2} R A (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} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R A (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u2} A (AddMonoid.toAddZeroClass.{u2} A (AddCommMonoid.toAddMonoid.{u2} A (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))))))) (Module.toMulActionWithZero.{u1, u2} R A (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} A (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} A (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))))) (Algebra.toModule.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8))))) (SMulZeroClass.toHasSmul.{u2, u3} A B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u2, u3} A B (MulZeroClass.toHasZero.{u2} A (MulZeroOneClass.toMulZeroClass.{u2} A (MonoidWithZero.toMulZeroOneClass.{u2} A (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u2, u3} A B (Semiring.toMonoidWithZero.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u2, u3} A B (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u2, u3} A B (CommRing.toCommSemiring.{u2} A _inst_2) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_6))))) (SMulZeroClass.toHasSmul.{u1, u3} R B (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (SMulWithZero.toSmulZeroClass.{u1, u3} R B (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.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (MulActionWithZero.toSMulWithZero.{u1, u3} R B (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (AddZeroClass.toHasZero.{u3} B (AddMonoid.toAddZeroClass.{u3} B (AddCommMonoid.toAddMonoid.{u3} B (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)))))))) (Module.toMulActionWithZero.{u1, u3} R B (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} B (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} B (Semiring.toNonAssocSemiring.{u3} B (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3))))) (Algebra.toModule.{u1, u3} R B (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u3} B (CommRing.toRing.{u3} B _inst_3)) _inst_7)))))] {x : B}, (IsIntegral.{u1, u3} R B _inst_1 (CommRing.toRing.{u3} B _inst_3) _inst_7 x) -> (IsIntegral.{u2, u3} A B _inst_2 (CommRing.toRing.{u3} B _inst_3) _inst_6 x)
+but is expected to have type
+  forall {R : Type.{u3}} {A : Type.{u1}} {B : Type.{u2}} [_inst_1 : CommRing.{u3} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : CommRing.{u2} B] [_inst_6 : Algebra.{u1, u2} A B (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_7 : Algebra.{u3, u2} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3))] [_inst_8 : Algebra.{u3, u1} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))] [_inst_9 : IsScalarTower.{u3, u1, u2} R A B (Algebra.toSMul.{u3, u1} R A (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_8) (Algebra.toSMul.{u1, u2} A B (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_6) (Algebra.toSMul.{u3, u2} R B (CommRing.toCommSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u2} B (CommRing.toCommSemiring.{u2} B _inst_3)) _inst_7)] {x : B}, (IsIntegral.{u3, u2} R B _inst_1 (CommRing.toRing.{u2} B _inst_3) _inst_7 x) -> (IsIntegral.{u1, u2} A B _inst_2 (CommRing.toRing.{u2} B _inst_3) _inst_6 x)
+Case conversion may be inaccurate. Consider using '#align is_integral_tower_top_of_is_integral isIntegral_tower_top_of_isIntegralₓ'. -/
 /-- If `R → A → B` is an algebra tower,
 then if the entire tower is an integral extension so is `A → B`. -/
 theorem isIntegral_tower_top_of_isIntegral {x : B} (h : IsIntegral R x) : IsIntegral A x :=
@@ -1091,6 +1659,12 @@ theorem isIntegral_tower_top_of_isIntegral {x : B} (h : IsIntegral R x) : IsInte
   exact hp'
 #align is_integral_tower_top_of_is_integral isIntegral_tower_top_of_isIntegral
 
+/- warning: ring_hom.is_integral_quotient_of_is_integral -> RingHom.isIntegral_quotient_of_isIntegral is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] (f : RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) {I : Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4))}, (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f) -> (RingHom.IsIntegral.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) f I)) (HasQuotient.Quotient.{u2, u2} S (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4))) (Ideal.hasQuotient.{u2} S _inst_4) I) (Ideal.Quotient.commRing.{u1} R _inst_1 (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) f I)) (CommRing.toRing.{u2} (HasQuotient.Quotient.{u2, u2} S (Ideal.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4))) (Ideal.hasQuotient.{u2} S _inst_4) I) (Ideal.Quotient.commRing.{u2} S _inst_4 I)) (Ideal.quotientMap.{u1, u2} R S _inst_1 _inst_4 (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) f I) I f (le_rfl.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) (SetLike.partialOrder.{u1, u1} (Ideal.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))) R (Submodule.setLike.{u1, u1} R R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)))))) (Ideal.comap.{u1, u2, max u1 u2} R S (RingHom.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_4)) (RingHom.ringHomClass.{u1, u2} R S (NonAssocRing.toNonAssocSemiring.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (NonAssocRing.toNonAssocSemiring.{u2} S (Ring.toNonAssocRing.{u2} S (CommRing.toRing.{u2} S _inst_4)))) f I))))
+but is expected to have type
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] (f : 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_4)))) {I : Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))}, (RingHom.IsIntegral.{u1, u2} R S _inst_1 (CommRing.toRing.{u2} S _inst_4) f) -> (RingHom.IsIntegral.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Ideal.comap.{u1, u2, max u1 u2} R S (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_4)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)) (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_4)))) f I)) (HasQuotient.Quotient.{u2, u2} S (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} S _inst_4) I) (Ideal.Quotient.commRing.{u1} R _inst_1 (Ideal.comap.{u1, u2, max u1 u2} R S (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_4)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)) (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_4)))) f I)) (CommRing.toRing.{u2} (HasQuotient.Quotient.{u2, u2} S (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} S _inst_4) I) (Ideal.Quotient.commRing.{u2} S _inst_4 I)) (Ideal.quotientMap.{u1, u2} R S _inst_1 _inst_4 (Ideal.comap.{u1, u2, max u1 u2} R S (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_4)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)) (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_4)))) f I) I f (le_rfl.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))) (Semiring.toModule.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))))))) (Ideal.comap.{u1, u2, max u1 u2} R S (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_4)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)) (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_4)))) f I))))
+Case conversion may be inaccurate. Consider using '#align ring_hom.is_integral_quotient_of_is_integral RingHom.isIntegral_quotient_of_isIntegralₓ'. -/
 theorem RingHom.isIntegral_quotient_of_isIntegral {I : Ideal S} (hf : f.IsIntegral) :
     (Ideal.quotientMap I f le_rfl).IsIntegral :=
   by
@@ -1100,11 +1674,23 @@ theorem RingHom.isIntegral_quotient_of_isIntegral {I : Ideal S} (hf : f.IsIntegr
   simpa only [hom_eval₂, eval₂_map] using congr_arg (Ideal.Quotient.mk I) hpx
 #align ring_hom.is_integral_quotient_of_is_integral RingHom.isIntegral_quotient_of_isIntegral
 
+/- warning: is_integral_quotient_of_is_integral -> isIntegral_quotient_of_isIntegral is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_8 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))] {I : Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))}, (Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8) -> (Algebra.IsIntegral.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.hasQuotient.{u1} R _inst_1) (Ideal.comap.{u1, u2, max u1 u2} R A (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (RingHom.ringHomClass.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))) (algebraMap.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8) I)) (HasQuotient.Quotient.{u2, u2} A (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Ideal.hasQuotient.{u2} A _inst_2) I) (Ideal.Quotient.commRing.{u1} R _inst_1 (Ideal.comap.{u1, u2, max u1 u2} R A (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) (RingHom.ringHomClass.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)))) (algebraMap.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2)) _inst_8) I)) (CommRing.toRing.{u2} (HasQuotient.Quotient.{u2, u2} A (Ideal.{u2} A (Ring.toSemiring.{u2} A (CommRing.toRing.{u2} A _inst_2))) (Ideal.hasQuotient.{u2} A _inst_2) I) (Ideal.Quotient.commRing.{u2} A _inst_2 I)) (Ideal.quotientAlgebra.{u1, u2} R _inst_1 A _inst_2 I _inst_8))
+but is expected to have type
+  forall {R : Type.{u1}} {A : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} A] [_inst_8 : Algebra.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))] {I : Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))}, (Algebra.IsIntegral.{u1, u2} R A _inst_1 (CommRing.toRing.{u2} A _inst_2) _inst_8) -> (Algebra.IsIntegral.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Ideal.comap.{u1, u2, max u1 u2} R A (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (RingHom.instRingHomClassRingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (algebraMap.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_8) I)) (HasQuotient.Quotient.{u2, u2} A (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} A _inst_2) I) (Ideal.Quotient.commRing.{u1} R _inst_1 (Ideal.comap.{u1, u2, max u1 u2} R A (RingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) (RingHom.instRingHomClassRingHom.{u1, u2} R A (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Semiring.toNonAssocSemiring.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)))) (algebraMap.{u1, u2} R A (CommRing.toCommSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2)) _inst_8) I)) (CommRing.toRing.{u2} (HasQuotient.Quotient.{u2, u2} A (Ideal.{u2} A (CommSemiring.toSemiring.{u2} A (CommRing.toCommSemiring.{u2} A _inst_2))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} A _inst_2) I) (Ideal.Quotient.commRing.{u2} A _inst_2 I)) (Ideal.quotientAlgebra.{u1, u2} R _inst_1 A _inst_2 I _inst_8))
+Case conversion may be inaccurate. Consider using '#align is_integral_quotient_of_is_integral isIntegral_quotient_of_isIntegralₓ'. -/
 theorem isIntegral_quotient_of_isIntegral {I : Ideal A} (hRA : Algebra.IsIntegral R A) :
     Algebra.IsIntegral (R ⧸ I.comap (algebraMap R A)) (A ⧸ I) :=
   (algebraMap R A).isIntegral_quotient_of_isIntegral hRA
 #align is_integral_quotient_of_is_integral isIntegral_quotient_of_isIntegral
 
+/- warning: is_integral_quotient_map_iff -> isIntegral_quotientMap_iff is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_4 : CommRing.{u2} S] (f : 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_4)))) {I : Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))}, Iff (RingHom.IsIntegral.{u1, u2} (HasQuotient.Quotient.{u1, u1} R (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u1} R _inst_1) (Ideal.comap.{u1, u2, max u1 u2} R S (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_4)))) (CommSemiring.toSemiring.{u1} R 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R _inst_1)) (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4)) (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_4)))) f I) I f (le_rfl.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (PartialOrder.toPreorder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Ideal.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1))) (Submodule.completeLattice.{u1, u1} R R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) 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(CommRing.toCommSemiring.{u2} S _inst_4)))) f I)))) (RingHom.IsIntegral.{u1, u2} R (HasQuotient.Quotient.{u2, u2} S (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} S _inst_4) I) _inst_1 (CommRing.toRing.{u2} (HasQuotient.Quotient.{u2, u2} S (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} S _inst_4) I) (Ideal.Quotient.commRing.{u2} S _inst_4 I)) (RingHom.comp.{u1, u2, u2} R S (HasQuotient.Quotient.{u2, u2} S (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} S _inst_4) I) (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_4))) (Semiring.toNonAssocSemiring.{u2} (HasQuotient.Quotient.{u2, u2} S (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} S _inst_4) I) (CommSemiring.toSemiring.{u2} (HasQuotient.Quotient.{u2, u2} S (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} S _inst_4) I) (CommRing.toCommSemiring.{u2} (HasQuotient.Quotient.{u2, u2} S (Ideal.{u2} S (CommSemiring.toSemiring.{u2} S (CommRing.toCommSemiring.{u2} S _inst_4))) (Ideal.instHasQuotientIdealToSemiringToCommSemiring.{u2} S _inst_4) I) (Ideal.Quotient.commRing.{u2} S _inst_4 I)))) (Ideal.Quotient.mk.{u2} S _inst_4 I) f))
+Case conversion may be inaccurate. Consider using '#align is_integral_quotient_map_iff isIntegral_quotientMap_iffₓ'. -/
 theorem isIntegral_quotientMap_iff {I : Ideal S} :
     (Ideal.quotientMap I f le_rfl).IsIntegral ↔
       ((Ideal.Quotient.mk I).comp f : R →+* S ⧸ I).IsIntegral :=
@@ -1116,6 +1702,12 @@ theorem isIntegral_quotientMap_iff {I : Ideal S} :
   exact RingHom.isIntegral_of_surjective g Ideal.Quotient.mk_surjective
 #align is_integral_quotient_map_iff isIntegral_quotientMap_iff
 
+/- warning: is_field_of_is_integral_of_is_field -> isField_of_isIntegral_of_isField is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_10 : CommRing.{u1} R] [_inst_11 : Nontrivial.{u1} R] [_inst_12 : CommRing.{u2} S] [_inst_13 : IsDomain.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_12))] [_inst_14 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_10) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_12))], (Algebra.IsIntegral.{u1, u2} R S _inst_10 (CommRing.toRing.{u2} S _inst_12) _inst_14) -> (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_10))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_12)))) (fun (_x : RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_10))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_12)))) => R -> S) (RingHom.hasCoeToFun.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_10))) (Semiring.toNonAssocSemiring.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_12)))) (algebraMap.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_10) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_12)) _inst_14))) -> (IsField.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_12))) -> (IsField.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_10)))
+but is expected to have type
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+Case conversion may be inaccurate. Consider using '#align is_field_of_is_integral_of_is_field isField_of_isIntegral_of_isFieldₓ'. -/
 /-- If the integral extension `R → S` is injective, and `S` is a field, then `R` is also a field. -/
 theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial R] [CommRing S]
     [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S)
@@ -1156,14 +1748,20 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
   rw [mul_assoc, ← pow_succ', tsub_add_cancel_of_le this]
 #align is_field_of_is_integral_of_is_field isField_of_isIntegral_of_isField
 
-theorem isField_of_isIntegral_of_is_field' {R S : Type _} [CommRing R] [CommRing S] [IsDomain S]
+/- warning: is_field_of_is_integral_of_is_field' -> isField_of_isIntegral_of_isField' is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_10 : CommRing.{u1} R] [_inst_11 : CommRing.{u2} S] [_inst_12 : IsDomain.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_11))] [_inst_13 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_10) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_11))], (Algebra.IsIntegral.{u1, u2} R S _inst_10 (CommRing.toRing.{u2} S _inst_11) _inst_13) -> (IsField.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_10))) -> (IsField.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_11)))
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_10 : CommRing.{u2} R] [_inst_11 : CommRing.{u1} S] [_inst_12 : IsDomain.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_11))] [_inst_13 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_10) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_11))], (Algebra.IsIntegral.{u2, u1} R S _inst_10 (CommRing.toRing.{u1} S _inst_11) _inst_13) -> (IsField.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_10))) -> (IsField.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_11)))
+Case conversion may be inaccurate. Consider using '#align is_field_of_is_integral_of_is_field' isField_of_isIntegral_of_isField'ₓ'. -/
+theorem isField_of_isIntegral_of_isField' {R S : Type _} [CommRing R] [CommRing S] [IsDomain S]
     [Algebra R S] (H : Algebra.IsIntegral R S) (hR : IsField R) : IsField S :=
   by
   letI := hR.to_field
   refine' ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun x hx => _⟩
   let A := Algebra.adjoin R ({x} : Set S)
   haveI : IsNoetherian R A :=
-    isNoetherian_of_fg_of_noetherian A.to_submodule (fG_adjoin_singleton_of_integral x (H x))
+    isNoetherian_of_fg_of_noetherian A.to_submodule (FG_adjoin_singleton_of_integral x (H x))
   haveI : Module.Finite R A := Module.IsNoetherian.finite R A
   obtain ⟨y, hy⟩ :=
     LinearMap.surjective_of_injective
@@ -1171,16 +1769,28 @@ theorem isField_of_isIntegral_of_is_field' {R S : Type _} [CommRing R] [CommRing
         hx (subtype.ext_iff.mp h))
       1
   exact ⟨y, subtype.ext_iff.mp hy⟩
-#align is_field_of_is_integral_of_is_field' isField_of_isIntegral_of_is_field'
-
+#align is_field_of_is_integral_of_is_field' isField_of_isIntegral_of_isField'
+
+/- warning: algebra.is_integral.is_field_iff_is_field -> Algebra.IsIntegral.isField_iff_isField is a dubious translation:
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+Case conversion may be inaccurate. Consider using '#align algebra.is_integral.is_field_iff_is_field Algebra.IsIntegral.isField_iff_isFieldₓ'. -/
 theorem Algebra.IsIntegral.isField_iff_isField {R S : Type _} [CommRing R] [Nontrivial R]
     [CommRing S] [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S)
     (hRS : Function.Injective (algebraMap R S)) : IsField R ↔ IsField S :=
-  ⟨isField_of_isIntegral_of_is_field' H, isField_of_isIntegral_of_isField H hRS⟩
+  ⟨isField_of_isIntegral_of_isField' H, isField_of_isIntegral_of_isField H hRS⟩
 #align algebra.is_integral.is_field_iff_is_field Algebra.IsIntegral.isField_iff_isField
 
 end Algebra
 
+/- warning: integral_closure_idem -> integralClosure_idem is a dubious translation:
+lean 3 declaration is
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+but is expected to have type
+  forall {R : Type.{u2}} {A : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} A] [_inst_3 : Algebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2))], Eq.{succ u1} (Subalgebra.{u1, u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) A (CommRing.toCommSemiring.{u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (Subalgebra.toCommRing.{u2, u1} R A _inst_1 _inst_2 _inst_3 (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Subalgebra.toAlgebra.{u1, u2, u1} A R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 (Algebra.id.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (integralClosure.{u1, u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) A (Subalgebra.toCommRing.{u2, u1} R A _inst_1 _inst_2 _inst_3 (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3)) _inst_2 (Subalgebra.toAlgebra.{u1, u2, u1} A R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 (Algebra.id.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (Bot.bot.{u1} (Subalgebra.{u1, u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) A (CommRing.toCommSemiring.{u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (Subalgebra.toCommRing.{u2, u1} R A _inst_1 _inst_2 _inst_3 (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Subalgebra.toAlgebra.{u1, u2, u1} A R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 (Algebra.id.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (CompleteLattice.toBot.{u1} (Subalgebra.{u1, u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) A (CommRing.toCommSemiring.{u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (Subalgebra.toCommRing.{u2, u1} R A _inst_1 _inst_2 _inst_3 (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Subalgebra.toAlgebra.{u1, u2, u1} A R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 (Algebra.id.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (Algebra.instCompleteLatticeSubalgebra.{u1, u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) A (CommRing.toCommSemiring.{u1} (Set.Elem.{u1} A (SetLike.coe.{u1, u1} (Subalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) A (Subalgebra.instSetLikeSubalgebra.{u2, u1} R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (Subalgebra.toCommRing.{u2, u1} R A _inst_1 _inst_2 _inst_3 (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3))) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (Subalgebra.toAlgebra.{u1, u2, u1} A R A (CommRing.toCommSemiring.{u2} R _inst_1) (CommRing.toCommSemiring.{u1} A _inst_2) (CommSemiring.toSemiring.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) _inst_3 (Algebra.id.{u1} A (CommRing.toCommSemiring.{u1} A _inst_2)) (integralClosure.{u2, u1} R A _inst_1 _inst_2 _inst_3)))))
+Case conversion may be inaccurate. Consider using '#align integral_closure_idem integralClosure_idemₓ'. -/
 theorem integralClosure_idem {R : Type _} {A : Type _} [CommRing R] [CommRing A] [Algebra R A] :
     integralClosure (integralClosure R A : Set A) A = ⊥ :=
   eq_bot_iff.2 fun x hx =>
@@ -1198,6 +1808,12 @@ variable {R S : Type _} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S]
 instance : IsDomain (integralClosure R S) :=
   inferInstance
 
+/- warning: roots_mem_integral_closure -> roots_mem_integralClosure is a dubious translation:
+lean 3 declaration is
+  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : CommRing.{u1} R] [_inst_2 : CommRing.{u2} S] [_inst_3 : IsDomain.{u2} S (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] [_inst_4 : Algebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2))] {f : Polynomial.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1))}, (Polynomial.Monic.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R _inst_1)) f) -> (forall {a : S}, (Membership.Mem.{u2, u2} S (Multiset.{u2} S) (Multiset.hasMem.{u2} S) a (Polynomial.roots.{u2} S _inst_2 _inst_3 (Polynomial.map.{u1, u2} R S (CommSemiring.toSemiring.{u1} R (CommRing.toCommSemiring.{u1} R _inst_1)) (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_4) f))) -> (Membership.Mem.{u2, u2} S (Subalgebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_4) (SetLike.hasMem.{u2, u2} (Subalgebra.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_4) S (Subalgebra.setLike.{u1, u2} R S (CommRing.toCommSemiring.{u1} R _inst_1) (Ring.toSemiring.{u2} S (CommRing.toRing.{u2} S _inst_2)) _inst_4)) a (integralClosure.{u1, u2} R S _inst_1 _inst_2 _inst_4)))
+but is expected to have type
+  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : CommRing.{u2} R] [_inst_2 : CommRing.{u1} S] [_inst_3 : IsDomain.{u1} S (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] [_inst_4 : Algebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2))] {f : Polynomial.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1))}, (Polynomial.Monic.{u2} R (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) f) -> (forall {a : S}, (Membership.mem.{u1, u1} S (Multiset.{u1} S) (Multiset.instMembershipMultiset.{u1} S) a (Polynomial.roots.{u1} S _inst_2 _inst_3 (Polynomial.map.{u2, u1} R S (CommSemiring.toSemiring.{u2} R (CommRing.toCommSemiring.{u2} R _inst_1)) (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_4) f))) -> (Membership.mem.{u1, u1} S (Subalgebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_4) (SetLike.instMembership.{u1, u1} (Subalgebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_4) S (Subalgebra.instSetLikeSubalgebra.{u2, u1} R S (CommRing.toCommSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u1} S (CommRing.toCommSemiring.{u1} S _inst_2)) _inst_4)) a (integralClosure.{u2, u1} R S _inst_1 _inst_2 _inst_4)))
+Case conversion may be inaccurate. Consider using '#align roots_mem_integral_closure roots_mem_integralClosureₓ'. -/
 theorem roots_mem_integralClosure {f : R[X]} (hf : f.Monic) {a : S}
     (ha : a ∈ (f.map <| algebraMap R S).roots) : a ∈ integralClosure R S :=
   ⟨f, hf, (eval₂_eq_eval_map _).trans <| (mem_roots <| (hf.map _).NeZero).1 ha⟩

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

@@ -208,8 +208,8 @@ theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
   exact isIntegral_ofSubring _ hsr
 #align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finite
 
-theorem fg_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
-    (Algebra.adjoin R ({x} : Set A)).toSubmodule.Fg :=
+theorem fG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
+    (Algebra.adjoin R ({x} : Set A)).toSubmodule.FG :=
   by
   rcases hx with ⟨f, hfm, hfx⟩
   exists Finset.image ((· ^ ·) x) (Finset.range (nat_degree f + 1))
@@ -237,10 +237,10 @@ theorem fg_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rw [degree_le_iff_coeff_zero] at this
   rw [mem_support_iff] at hkq; apply hkq; apply this
   exact lt_of_le_of_lt degree_le_nat_degree (WithBot.coe_lt_coe.2 hk)
-#align fg_adjoin_singleton_of_integral fg_adjoin_singleton_of_integral
+#align fg_adjoin_singleton_of_integral fG_adjoin_singleton_of_integral
 
-theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
-    (Algebra.adjoin R s).toSubmodule.Fg :=
+theorem fG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
+    (Algebra.adjoin R s).toSubmodule.FG :=
   Set.Finite.induction_on hfs
     (fun _ =>
       ⟨{1},
@@ -253,19 +253,19 @@ theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
       rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule] <;>
         exact
           fg.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)
-            (fg_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s))
+            (fG_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s))
     his
-#align fg_adjoin_of_finite fg_adjoin_of_finite
+#align fg_adjoin_of_finite fG_adjoin_of_finite
 
 theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
     (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (↑s : Set A)) :=
-  isNoetherian_of_fg_of_noetherian _ (fg_adjoin_of_finite s.finite_toSet hs)
+  isNoetherian_of_fg_of_noetherian _ (fG_adjoin_of_finite s.finite_toSet hs)
 #align is_noetherian_adjoin_finset isNoetherian_adjoin_finset
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
   then all elements of `S` are integral over `R`. -/
-theorem isIntegral_of_mem_of_fg (S : Subalgebra R A) (HS : S.toSubmodule.Fg) (x : A) (hx : x ∈ S) :
+theorem isIntegral_of_mem_of_fG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) :
     IsIntegral R x :=
   by
   -- say `x ∈ S`. We want to prove that `x` is integral over `R`.
@@ -362,7 +362,7 @@ theorem isIntegral_of_mem_of_fg (S : Subalgebra R A) (HS : S.toSubmodule.Fg) (x
   change (⟨_, this⟩ : S₀) • r ∈ _
   rw [Finsupp.mem_supported] at hlx1
   exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _
-#align is_integral_of_mem_of_fg isIntegral_of_mem_of_fg
+#align is_integral_of_mem_of_fg isIntegral_of_mem_of_fG
 
 theorem Module.End.isIntegral {M : Type _} [AddCommGroup M] [Module R M] [Module.Finite R M] :
     Algebra.IsIntegral R (Module.End R M) :=
@@ -372,7 +372,7 @@ theorem Module.End.isIntegral {M : Type _} [AddCommGroup M] [Module R M] [Module
 /-- Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
 and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. -/
 theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R M] [Module A M]
-    [IsScalarTower R A M] [NoZeroSMulDivisors A M] (N : Submodule R M) (hN : N ≠ ⊥) (hN' : N.Fg)
+    [IsScalarTower R A M] [NoZeroSMulDivisors A M] (N : Submodule R M) (hN : N ≠ ⊥) (hN' : N.FG)
     (x : A) (hx : ∀ n ∈ N, x • n ∈ N) : IsIntegral R x :=
   by
   let A' : Subalgebra R A :=
@@ -412,7 +412,7 @@ variable {f}
 
 theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
   letI := f.to_algebra
-  fun x => isIntegral_of_mem_of_fg ⊤ h.1 _ trivial
+  fun x => isIntegral_of_mem_of_fG ⊤ h.1 _ trivial
 #align ring_hom.finite.to_is_integral RingHom.Finite.to_isIntegral
 
 alias RingHom.Finite.to_isIntegral ← RingHom.IsIntegral.of_finite
@@ -423,9 +423,9 @@ theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.
   letI := f.to_algebra
   obtain ⟨s, hs⟩ := h'
   constructor
-  change (⊤ : Subalgebra R S).toSubmodule.Fg
+  change (⊤ : Subalgebra R S).toSubmodule.FG
   rw [← hs]
-  exact fg_adjoin_of_finite (Set.toFinite _) fun x _ => h x
+  exact fG_adjoin_of_finite (Set.toFinite _) fun x _ => h x
 #align ring_hom.is_integral.to_finite RingHom.IsIntegral.to_finite
 
 alias RingHom.IsIntegral.to_finite ← RingHom.Finite.of_isIntegral_of_finiteType
@@ -467,10 +467,10 @@ theorem RingHom.is_integral_of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
     (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z :=
   by
   letI : Algebra R S := f.to_algebra
-  have := (fg_adjoin_singleton_of_integral x hx).mul (fg_adjoin_singleton_of_integral y hy)
+  have := (fG_adjoin_singleton_of_integral x hx).mul (fG_adjoin_singleton_of_integral y hy)
   rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this
   exact
-    isIntegral_of_mem_of_fg (Algebra.adjoin R {x, y}) this z
+    isIntegral_of_mem_of_fG (Algebra.adjoin R {x, y}) this z
       (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)
 #align ring_hom.is_integral_of_mem_closure RingHom.is_integral_of_mem_closure
 
@@ -563,12 +563,12 @@ def integralClosure : Subalgebra R A
   algebraMap_mem' x := isIntegral_algebraMap
 #align integral_closure integralClosure
 
-theorem mem_integralClosure_iff_mem_fg {r : A} :
-    r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.Fg ∧ r ∈ M :=
+theorem mem_integralClosure_iff_mem_fG {r : A} :
+    r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M :=
   ⟨fun hr =>
-    ⟨Algebra.adjoin R {r}, fg_adjoin_singleton_of_integral _ hr, Algebra.subset_adjoin rfl⟩,
-    fun ⟨M, Hf, hrM⟩ => isIntegral_of_mem_of_fg M Hf _ hrM⟩
-#align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_fg
+    ⟨Algebra.adjoin R {r}, fG_adjoin_singleton_of_integral _ hr, Algebra.subset_adjoin rfl⟩,
+    fun ⟨M, Hf, hrM⟩ => isIntegral_of_mem_of_fG M Hf _ hrM⟩
+#align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_fG
 
 variable {R} {A}
 
@@ -1011,13 +1011,13 @@ theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegr
     IsIntegral R x := by
   rcases hx with ⟨p, pmonic, hp⟩
   let S : Set B := ↑(p.map <| algebraMap A B).frange
-  refine' isIntegral_of_mem_of_fg (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
-  refine' fg_trans (fg_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
+  refine' isIntegral_of_mem_of_fG (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
+  refine' fg_trans (fG_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
   · rw [Finset.mem_coe, frange, Finset.mem_image] at hx
     rcases hx with ⟨i, _, rfl⟩
     rw [coeff_map]
     exact map_isIntegral (IsScalarTower.toAlgHom R A B) (A_int _)
-  · apply fg_adjoin_singleton_of_integral
+  · apply fG_adjoin_singleton_of_integral
     exact isIntegral_trans_aux _ pmonic hp
 #align is_integral_trans isIntegral_trans
 
@@ -1163,7 +1163,7 @@ theorem isField_of_isIntegral_of_is_field' {R S : Type _} [CommRing R] [CommRing
   refine' ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun x hx => _⟩
   let A := Algebra.adjoin R ({x} : Set S)
   haveI : IsNoetherian R A :=
-    isNoetherian_of_fg_of_noetherian A.to_submodule (fg_adjoin_singleton_of_integral x (H x))
+    isNoetherian_of_fg_of_noetherian A.to_submodule (fG_adjoin_singleton_of_integral x (H x))
   haveI : Module.Finite R A := Module.IsNoetherian.finite R A
   obtain ⟨y, hy⟩ :=
     LinearMap.surjective_of_injective

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

@@ -289,7 +289,7 @@ theorem isIntegral_of_mem_of_fg (S : Subalgebra R A) (HS : S.toSubmodule.Fg) (x
   choose ly hly1 hly2
   -- Now let `S₀` be the subring of `R` generated by the `rᵢ` and the `rᵢⱼₖ`.
   let S₀ : Subring R :=
-    Subring.closure ↑(lx.frange ∪ Finset.bunionᵢ Finset.univ (Finsupp.frange ∘ ly))
+    Subring.closure ↑(lx.frange ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))
   -- It suffices to prove that `x` is integral over `S₀`.
   refine' isIntegral_ofSubring S₀ _
   letI : CommRing S₀ := SubringClass.toCommRing S₀
@@ -313,7 +313,7 @@ theorem isIntegral_of_mem_of_fg (S : Subalgebra R A) (HS : S.toSubmodule.Fg) (x
     have : ly ⟨(p, q), Finset.mem_product.2 ⟨hp, hq⟩⟩ t ∈ S₀ :=
       Subring.subset_closure
         (Finset.mem_union_right _ <|
-          Finset.mem_bunionᵢ.2
+          Finset.mem_biUnion.2
             ⟨⟨(p, q), Finset.mem_product.2 ⟨hp, hq⟩⟩, Finset.mem_univ _,
               Finsupp.mem_frange.2 ⟨Finsupp.mem_support_iff.1 ht, _, rfl⟩⟩)
     change (⟨_, this⟩ : S₀) • t ∈ _

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

@@ -823,7 +823,7 @@ end
 
 section IsIntegralClosure
 
-/- ./././Mathport/Syntax/Translate/Command.lean:388:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
+/- ./././Mathport/Syntax/Translate/Command.lean:393:30: infer kinds are unsupported in Lean 4: #[`algebraMap_injective] [] -/
 /-- `is_integral_closure A R B` is the characteristic predicate stating `A` is
 the integral closure of `R` in `B`,
 i.e. that an element of `B` is integral over `R` iff it is an element of (the image of) `A`.

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

@@ -692,14 +692,13 @@ theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x 
   refine' ⟨(p.map (algebraMap R A)).scaleRoots x, _, _⟩
   · rw [Polynomial.monic_scaleRoots_iff]
     exact hp.map _
-  convert
-    @Polynomial.scaleRoots_eval₂_mul (A ⊗[R] B) A _ _ _
+  convert@Polynomial.scaleRoots_eval₂_mul (A ⊗[R] B) A _ _ _
       algebra.tensor_product.include_left.to_ring_hom (1 ⊗ₜ y) x using
     2
   ·
     simp only [AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom, mul_one, one_mul,
       Algebra.TensorProduct.includeLeft_apply, Algebra.TensorProduct.tmul_mul_tmul]
-  convert (MulZeroClass.mul_zero _).symm
+  convert(MulZeroClass.mul_zero _).symm
   rw [Polynomial.eval₂_map, Algebra.TensorProduct.includeLeft_comp_algebraMap, ←
     Polynomial.eval₂_map]
   convert Polynomial.eval₂_at_apply algebra.tensor_product.include_right.to_ring_hom y

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

@@ -225,7 +225,7 @@ theorem fg_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rcases(aeval x).mem_range.mp hr with ⟨p, rfl⟩
   rw [← mod_by_monic_add_div p hfm]
   rw [← aeval_def] at hfx
-  rw [AlgHom.map_add, AlgHom.map_mul, hfx, zero_mul, add_zero]
+  rw [AlgHom.map_add, AlgHom.map_mul, hfx, MulZeroClass.zero_mul, add_zero]
   have : degree (p %ₘ f) ≤ degree f := degree_mod_by_monic_le p hfm
   generalize p %ₘ f = q at this⊢
   rw [← sum_C_mul_X_pow_eq q, aeval_def, eval₂_sum, sum_def]
@@ -699,7 +699,7 @@ theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x 
   ·
     simp only [AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom, mul_one, one_mul,
       Algebra.TensorProduct.includeLeft_apply, Algebra.TensorProduct.tmul_mul_tmul]
-  convert (mul_zero _).symm
+  convert (MulZeroClass.mul_zero _).symm
   rw [Polynomial.eval₂_map, Algebra.TensorProduct.includeLeft_comp_algebraMap, ←
     Polynomial.eval₂_map]
   convert Polynomial.eval₂_at_apply algebra.tensor_product.include_right.to_ring_hom y
@@ -721,7 +721,7 @@ theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ nat
       p.coeff i * p.leadingCoeff ^ (p.natDegree - 1) :=
   by
   simp only [normalizeScaleRoots, finset_sum_coeff, coeff_monomial, Finset.sum_ite_eq', one_mul,
-    zero_mul, mem_support_iff, ite_mul, Ne.def, ite_not]
+    MulZeroClass.zero_mul, mem_support_iff, ite_mul, Ne.def, ite_not]
   split_ifs with h₁ h₂
   · simp [h₁]
   · rw [h₂, leading_coeff, ← pow_succ, tsub_add_cancel_of_le hp]
@@ -796,11 +796,11 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
       rw [h'', nat_degree_zero] at h'
       exact Nat.not_succ_le_zero 0 h'
     use normalizeScaleRoots_monic p this
-    rw [normalizeScaleRoots_eval₂_leadingCoeff_mul p h' f x, h, mul_zero]
+    rw [normalizeScaleRoots_eval₂_leadingCoeff_mul p h' f x, h, MulZeroClass.mul_zero]
   · by_cases hp : p.map f = 0
     · apply_fun fun q => coeff q p.nat_degree  at hp
       rw [coeff_map, coeff_zero, coeff_nat_degree] at hp
-      rw [hp, zero_mul]
+      rw [hp, MulZeroClass.zero_mul]
       exact f.is_integral_zero
     · rw [Nat.one_le_iff_ne_zero, Classical.not_not] at h'
       rw [eq_C_of_nat_degree_eq_zero h', eval₂_C] at h

mathlib commit https://github.com/leanprover-community/mathlib/commit/21e3562c5e12d846c7def5eff8cdbc520d7d4936

@@ -79,7 +79,7 @@ protected def Algebra.IsIntegral : Prop :=
 variable {R A}
 
 theorem RingHom.is_integral_map {x : R} : f.IsIntegralElem (f x) :=
-  ⟨X - C x, monic_x_sub_c _, by simp⟩
+  ⟨X - C x, monic_X_sub_C _, by simp⟩
 #align ring_hom.is_integral_map RingHom.is_integral_map
 
 theorem isIntegral_algebraMap {x : R} : IsIntegral R (algebraMap R A x) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -79,7 +79,7 @@ protected def Algebra.IsIntegral : Prop :=
 variable {R A}
 
 theorem RingHom.is_integral_map {x : R} : f.IsIntegralElem (f x) :=
-  ⟨x - c x, monic_x_sub_c _, by simp⟩
+  ⟨X - C x, monic_x_sub_c _, by simp⟩
 #align ring_hom.is_integral_map RingHom.is_integral_map
 
 theorem isIntegral_algebraMap {x : R} : IsIntegral R (algebraMap R A x) :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/641b6a82006416ec431b2987b354af9311fed4f2

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 
 ! This file was ported from Lean 3 source module ring_theory.integral_closure
-! leanprover-community/mathlib commit 825edd3cd735e87495b0c2a2114fc3929eefce41
+! leanprover-community/mathlib commit 641b6a82006416ec431b2987b354af9311fed4f2
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -860,6 +860,15 @@ theorem isIntegral_algebra [Algebra R A] [IsScalarTower R A B] : Algebra.IsInteg
   IsIntegralClosure.isIntegral R B x
 #align is_integral_closure.is_integral_algebra IsIntegralClosure.isIntegral_algebra
 
+theorem noZeroSMulDivisors [Algebra R A] [IsScalarTower R A B] [NoZeroSMulDivisors R B] :
+    NoZeroSMulDivisors R A :=
+  by
+  refine'
+    Function.Injective.noZeroSMulDivisors _ (IsIntegralClosure.algebraMap_injective A R B)
+      (map_zero _) fun _ _ => _
+  simp only [Algebra.algebraMap_eq_smul_one, IsScalarTower.smul_assoc]
+#align is_integral_closure.no_zero_smul_divisors IsIntegralClosure.noZeroSMulDivisors
+
 variable {R} (A) {B}
 
 /-- If `x : B` is integral over `R`, then it is an element of the integral closure of `R` in `B`. -/

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

This refactor adds a new definition IsIntegrallyClosedIn R A equal to IsIntegralClosure R A A, and redefines IsIntegrallyClosed R to equal IsIntegrallyClosed R (FractionRing A). This should make it possible and convenient to generalize away from the fraction fields.

This also more closely approximates the conventions of the Stacks project.

This is a second attempt at the refactor, after #7116 which was much more messy.

@@ -879,6 +879,16 @@ protected theorem RingHom.IsIntegral.trans
   Algebra.IsIntegral.trans hf hg
 #align ring_hom.is_integral_trans RingHom.IsIntegral.trans
 
+/-- If `R → A → B` is an algebra tower, `C` is the integral closure of `R` in `B`
+and `A` is integral over `R`, then `C` is the integral closure of `A` in `B`. -/
+lemma IsIntegralClosure.tower_top {B C : Type*} [CommRing C] [CommRing B]
+    [Algebra R B] [Algebra A B] [Algebra C B] [IsScalarTower R A B]
+    [IsIntegralClosure C R B] (hRA : Algebra.IsIntegral R A) :
+    IsIntegralClosure C A B :=
+  ⟨IsIntegralClosure.algebraMap_injective _ R _,
+   fun hx => (IsIntegralClosure.isIntegral_iff).mp (isIntegral_trans hRA _ hx),
+   fun hx => ((IsIntegralClosure.isIntegral_iff (R := R)).mpr hx).tower_top⟩
+
 theorem RingHom.isIntegral_of_surjective (hf : Function.Surjective f) : f.IsIntegral :=
   fun x ↦ (hf x).recOn fun _y hy ↦ hy ▸ f.isIntegralElem_map
 #align ring_hom.is_integral_of_surjective RingHom.isIntegral_of_surjective

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

@@ -840,7 +840,7 @@ variable [Algebra A B] [Algebra R B] (f : R →+* S) (g : S →+* T)
 variable [Algebra R A] [IsScalarTower R A B]
 
 /-- If A is an R-algebra all of whose elements are integral over R,
-and x is an element of an A-algebra that is integral over A, then x is integral over R.-/
+and x is an element of an A-algebra that is integral over A, then x is integral over R. -/
 theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegral A x) :
     IsIntegral R x := by
   rcases hx with ⟨p, pmonic, hp⟩
@@ -866,7 +866,7 @@ theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegr
 
 /-- If A is an R-algebra all of whose elements are integral over R,
 and B is an A-algebra all of whose elements are integral over A,
-then all elements of B are integral over R.-/
+then all elements of B are integral over R. -/
 protected theorem Algebra.IsIntegral.trans
     (hA : Algebra.IsIntegral R A) (hB : Algebra.IsIntegral A B) : Algebra.IsIntegral R B :=
   fun x ↦ isIntegral_trans hA x (hB x)

Polynomial and MvPolynomial are algebraic objects, hence should be under Algebra (or at least not under Data)

@@ -3,7 +3,7 @@ Copyright (c) 2019 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
 -/
-import Mathlib.Data.Polynomial.Expand
+import Mathlib.Algebra.Polynomial.Expand
 import Mathlib.LinearAlgebra.FiniteDimensional
 import Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap
 import Mathlib.RingTheory.Adjoin.FG

@@ -597,7 +597,7 @@ theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ nat
     (normalizeScaleRoots p).coeff i * p.leadingCoeff ^ i =
       p.coeff i * p.leadingCoeff ^ (p.natDegree - 1) := by
   simp only [normalizeScaleRoots, finset_sum_coeff, coeff_monomial, Finset.sum_ite_eq', one_mul,
-    zero_mul, mem_support_iff, ite_mul, Ne.def, ite_not]
+    zero_mul, mem_support_iff, ite_mul, Ne, ite_not]
   split_ifs with h₁ h₂
   · simp [h₁]
   · rw [h₂, leadingCoeff, ← pow_succ', tsub_add_cancel_of_le hp]
@@ -611,7 +611,7 @@ theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
     p.leadingCoeff • normalizeScaleRoots p = scaleRoots p p.leadingCoeff := by
   ext
   simp only [coeff_scaleRoots, normalizeScaleRoots, coeff_monomial, coeff_smul, Finset.smul_sum,
-    Ne.def, Finset.sum_ite_eq', finset_sum_coeff, smul_ite, smul_zero, mem_support_iff]
+    Ne, Finset.sum_ite_eq', finset_sum_coeff, smul_ite, smul_zero, mem_support_iff]
   -- Porting note: added the following `simp only`
   simp only [ge_iff_le, tsub_le_iff_right, smul_eq_mul, mul_ite, mul_one, mul_zero,
     Finset.sum_ite_eq', mem_support_iff, ne_eq, ite_not]
@@ -628,7 +628,7 @@ theorem normalizeScaleRoots_support : (normalizeScaleRoots p).support ≤ p.supp
   intro x
   contrapose
   simp only [not_mem_support_iff, normalizeScaleRoots, finset_sum_coeff, coeff_monomial,
-    Finset.sum_ite_eq', mem_support_iff, Ne.def, Classical.not_not, ite_eq_right_iff]
+    Finset.sum_ite_eq', mem_support_iff, Ne, Classical.not_not, ite_eq_right_iff]
   intro h₁ h₂
   exact (h₂ h₁).elim
 #align normalize_scale_roots_support normalizeScaleRoots_support

We change the following field in the definition of an additive commutative monoid:

 nsmul_succ : ∀ (n : ℕ) (x : G),
-  AddMonoid.nsmul (n + 1) x = x + AddMonoid.nsmul n x
+  AddMonoid.nsmul (n + 1) x = AddMonoid.nsmul n x + x

where the latter is more natural

We adjust the definitions of ^ in monoids, groups, etc. Originally there was a warning comment about why this natural order was preferred

use x * npowRec n x and not npowRec n x * x in the definition to make sure that definitional unfolding of npowRec is blocked, to avoid deep recursion issues.

but it seems to no longer apply.

Remarks on the PR :

• pow_succ and pow_succ' have switched their meanings.
• Most of the time, the proofs were adjusted by priming/unpriming one lemma, or exchanging left and right; a few proofs were more complicated to adjust.
• In particular, [Mathlib/NumberTheory/RamificationInertia.lean] used Ideal.IsPrime.mul_mem_pow which is defined in [Mathlib/RingTheory/DedekindDomain/Ideal.lean]. Changing the order of operation forced me to add the symmetric lemma Ideal.IsPrime.mem_pow_mul.
• the docstring for Cauchy condensation test in [Mathlib/Analysis/PSeries.lean] was mathematically incorrect, I added the mention that the function is antitone.

@@ -600,7 +600,7 @@ theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ nat
     zero_mul, mem_support_iff, ite_mul, Ne.def, ite_not]
   split_ifs with h₁ h₂
   · simp [h₁]
-  · rw [h₂, leadingCoeff, ← pow_succ, tsub_add_cancel_of_le hp]
+  · rw [h₂, leadingCoeff, ← pow_succ', tsub_add_cancel_of_le hp]
   · rw [mul_assoc, ← pow_add, tsub_add_cancel_of_le]
     apply Nat.le_sub_one_of_lt
     rw [lt_iff_le_and_ne]
@@ -618,7 +618,7 @@ theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
   split_ifs with h₁ h₂
   · simp [*]
   · simp [*]
-  · rw [mul_comm, mul_assoc, ← pow_succ', tsub_right_comm,
+  · rw [mul_comm, mul_assoc, ← pow_succ, tsub_right_comm,
       tsub_add_cancel_of_le]
     rw [Nat.succ_le_iff]
     exact tsub_pos_of_lt (lt_of_le_of_ne (le_natDegree_of_ne_zero h₁) h₂)

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)

@@ -39,7 +39,6 @@ open BigOperators Polynomial Submodule
 section Ring
 
 variable {R S A : Type*}
-
 variable [CommRing R] [Ring A] [Ring S] (f : R →+* S)
 
 /-- An element `x` of `A` is said to be integral over `R` with respect to `f`
@@ -85,9 +84,7 @@ end Ring
 section
 
 variable {R A B S : Type*}
-
 variable [CommRing R] [CommRing A] [Ring B] [CommRing S]
-
 variable [Algebra R A] [Algebra R B] (f : R →+* S)
 
 theorem IsIntegral.map {B C F : Type*} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
@@ -723,9 +720,7 @@ theorem algebraMap_injective (A R B : Type*) [CommRing R] [CommSemiring A] [Comm
   algebraMap_injective' R
 
 variable {R A B : Type*} [CommRing R] [CommRing A] [CommRing B]
-
 variable [Algebra R B] [Algebra A B] [IsIntegralClosure A R B]
-
 variable (R B)
 
 protected theorem isIntegral [Algebra R A] [IsScalarTower R A B] (x : A) : IsIntegral R x :=
@@ -791,7 +786,6 @@ section lift
 variable (B) {S : Type*} [CommRing S] [Algebra R S]
 -- split from above, since otherwise it does not synthesize `Semiring S`
 variable [Algebra S B] [IsScalarTower R S B]
-
 variable [Algebra R A] [IsScalarTower R A B] (h : Algebra.IsIntegral R S)
 
 /-- If `B / S / R` is a tower of ring extensions where `S` is integral over `R`,
@@ -816,7 +810,6 @@ section Equiv
 
 variable (R B) (A' : Type*) [CommRing A']
 variable [Algebra A' B] [IsIntegralClosure A' R B]
-
 variable [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A' B]
 
 /-- Integral closures are all isomorphic to each other. -/
@@ -842,11 +835,8 @@ section Algebra
 open Algebra
 
 variable {R A B S T : Type*}
-
 variable [CommRing R] [CommRing A] [Ring B] [CommRing S] [CommRing T]
-
 variable [Algebra A B] [Algebra R B] (f : R →+* S) (g : S →+* T)
-
 variable [Algebra R A] [IsScalarTower R A B]
 
 /-- If A is an R-algebra all of whose elements are integral over R,

Move:

• Mathlib/Algebra/Module/DirectLimitAndTensorProduct.lean to LinearAlgebra/TensorProduct/DirectLimit.lean
• Mathlib/LinearAlgebra/TensorProduct to Mathlib/LinearAlgebra.TensorProduct.Basic.lean
• Mathlib/RingTheory/TensorProduct to Mathlib/RingTheory/TensorProduct/Basic.lean.

This follows suggestions 1, 2, 3 of

https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Tensor.20Products.20of.20modules.20and.20rings/near/424605543

Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr>

@@ -10,7 +10,7 @@ import Mathlib.RingTheory.Adjoin.FG
 import Mathlib.RingTheory.FiniteType
 import Mathlib.RingTheory.Polynomial.ScaleRoots
 import Mathlib.RingTheory.Polynomial.Tower
-import Mathlib.RingTheory.TensorProduct
+import Mathlib.RingTheory.TensorProduct.Basic
 
 #align_import ring_theory.integral_closure from "leanprover-community/mathlib"@"641b6a82006416ec431b2987b354af9311fed4f2"
 

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

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

@@ -33,7 +33,8 @@ Let `R` be a `CommRing` and let `A` be an R-algebra.
 -/
 
 
-open Classical BigOperators Polynomial Submodule
+open scoped Classical
+open BigOperators Polynomial Submodule
 
 section Ring
 

Homogenises porting notes via capitalisation and addition of whitespace.

It makes the following changes:

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

@@ -264,13 +264,13 @@ theorem isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R
   let f : A' →ₐ[R] Module.End R N :=
     AlgHom.ofLinearMap
       { toFun := fun x => (DistribMulAction.toLinearMap R M x).restrict x.prop
-        -- porting note: was
+        -- Porting note: was
                 -- `fun x y => LinearMap.ext fun n => Subtype.ext <| add_smul x y n`
         map_add' := by intros x y; ext; exact add_smul _ _ _
-        -- porting note: was
+        -- Porting note: was
                 --  `fun r s => LinearMap.ext fun n => Subtype.ext <| smul_assoc r s n`
         map_smul' := by intros r s; ext; apply smul_assoc }
-      -- porting note: the next two lines were
+      -- Porting note: the next two lines were
       --`(LinearMap.ext fun n => Subtype.ext <| one_smul _ _) fun x y =>`
       --`LinearMap.ext fun n => Subtype.ext <| mul_smul x y n`
       (by ext; apply one_smul)
@@ -614,7 +614,7 @@ theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
   ext
   simp only [coeff_scaleRoots, normalizeScaleRoots, coeff_monomial, coeff_smul, Finset.smul_sum,
     Ne.def, Finset.sum_ite_eq', finset_sum_coeff, smul_ite, smul_zero, mem_support_iff]
-  -- porting note: added the following `simp only`
+  -- Porting note: added the following `simp only`
   simp only [ge_iff_le, tsub_le_iff_right, smul_eq_mul, mul_ite, mul_one, mul_zero,
     Finset.sum_ite_eq', mem_support_iff, ne_eq, ite_not]
   split_ifs with h₁ h₂

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>

@@ -90,7 +90,8 @@ variable [CommRing R] [CommRing A] [Ring B] [CommRing S]
 variable [Algebra R A] [Algebra R B] (f : R →+* S)
 
 theorem IsIntegral.map {B C F : Type*} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
-    [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B} [AlgHomClass F A B C] (f : F)
+    [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B}
+    [FunLike F B C] [AlgHomClass F A B C] (f : F)
     (hb : IsIntegral R b) : IsIntegral R (f b) := by
   obtain ⟨P, hP⟩ := hb
   refine' ⟨P, hP.1, _⟩
@@ -138,7 +139,8 @@ theorem IsIntegral.tower_top [Algebra A B] [IsScalarTower R A B] {x : B}
 #align is_integral_of_is_scalar_tower IsIntegral.tower_top
 #align is_integral_tower_top_of_is_integral IsIntegral.tower_top
 
-theorem map_isIntegral_int {B C F : Type*} [Ring B] [Ring C] {b : B} [RingHomClass F B C] (f : F)
+theorem map_isIntegral_int {B C F : Type*} [Ring B] [Ring C] {b : B}
+    [FunLike F B C] [RingHomClass F B C] (f : F)
     (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
   hb.map (f : B →+* C).toIntAlgHom
 #align map_is_integral_int map_isIntegral_int

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

@@ -170,22 +170,29 @@ theorem isIntegral_iff_isIntegral_closure_finite {r : B} :
   exact hsr.of_subring _
 #align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finite
 
-theorem IsIntegral.fg_adjoin_singleton {x : B} (hx : IsIntegral R x) :
-    (Algebra.adjoin R {x}).toSubmodule.FG := by
-  rcases hx with ⟨f, hfm, hfx⟩
-  use (Finset.range <| f.natDegree + 1).image (x ^ ·)
+theorem Submodule.span_range_natDegree_eq_adjoin {R A} [CommRing R] [Semiring A] [Algebra R A]
+    {x : A} {f : R[X]} (hf : f.Monic) (hfx : aeval x f = 0) :
+    span R (Finset.image (x ^ ·) (Finset.range (natDegree f))) =
+      Subalgebra.toSubmodule (Algebra.adjoin R {x}) := by
+  nontriviality A
+  have hf1 : f ≠ 1 := by rintro rfl; simp [one_ne_zero' A] at hfx
   refine (span_le.mpr fun s hs ↦ ?_).antisymm fun r hr ↦ ?_
   · rcases Finset.mem_image.1 hs with ⟨k, -, rfl⟩
     exact (Algebra.adjoin R {x}).pow_mem (Algebra.subset_adjoin rfl) k
   rw [Subalgebra.mem_toSubmodule, Algebra.adjoin_singleton_eq_range_aeval] at hr
   rcases (aeval x).mem_range.mp hr with ⟨p, rfl⟩
-  rw [← modByMonic_add_div p hfm, map_add, map_mul, aeval_def x f, hfx,
+  rw [← modByMonic_add_div p hf, map_add, map_mul, hfx,
       zero_mul, add_zero, ← sum_C_mul_X_pow_eq (p %ₘ f), aeval_def, eval₂_sum, sum_def]
   refine sum_mem fun k hkq ↦ ?_
   rw [C_mul_X_pow_eq_monomial, eval₂_monomial, ← Algebra.smul_def]
-  exact smul_mem _ _ (subset_span <| Finset.mem_image_of_mem _ <| Finset.mem_range_succ_iff.mpr <|
-    (le_natDegree_of_mem_supp _ hkq).trans <| natDegree_modByMonic_le p hfm)
-#align fg_adjoin_singleton_of_integral IsIntegral.fg_adjoin_singleton
+  exact smul_mem _ _ (subset_span <| Finset.mem_image_of_mem _ <| Finset.mem_range.mpr <|
+    (le_natDegree_of_mem_supp _ hkq).trans_lt <| natDegree_modByMonic_lt p hf hf1)
+
+theorem IsIntegral.fg_adjoin_singleton {x : B} (hx : IsIntegral R x) :
+    (Algebra.adjoin R {x}).toSubmodule.FG := by
+  rcases hx with ⟨f, hfm, hfx⟩
+  use (Finset.range <| f.natDegree).image (x ^ ·)
+  exact span_range_natDegree_eq_adjoin hfm (by rwa [aeval_def])
 
 theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
     (Algebra.adjoin R s).toSubmodule.FG :=

• generalize image_rootSet, adjoin_rootSet_eq_range and splits_comp_of_splits in Data/Polynomial/Splits and use the last one to golf splits_of_algHom, splits_of_isScalarTower (introduced in # 8609).

• add three new lemmas mem_range_x_of_minpoly_splits to simplify the construction of IntermediateField.algHomEquivAlgHomOfIsAlgClosed and Algebra.IsAlgebraic.algHomEquivAlgHomOfIsAlgClosed, remove the IsAlgClosed condition and rename. They could be moved to an earlier file but I refrain from doing that. (#find_home says it's already in the right place)

• golf primitive_element_iff_algHom_eq_of_eval from # 8609, using a new lemma IsIntegral.minpoly_splits_tower_top for the last step.

• make integralClosure_algEquiv_restrict (from # 8714) computable and rename to AlgEquiv.mapIntegralClosure to follow camelCase naming convention and enable dot notation.

Co-authored-by: Xavier-François Roblot <46200072+xroblot@users.noreply.github.com> Co-authored-by: Junyan Xu <junyanxu.math@gmail.com>

@@ -469,27 +469,25 @@ theorem integralClosure_map_algEquiv [Algebra R S] (f : A ≃ₐ[R] S) :
 
 /-- An `AlgHom` between two rings restrict to an `AlgHom` between the integral closures inside
 them. -/
-def integralClosure_algHom_restrict [Algebra R S] (f : A →ₐ[R] S) :
+def AlgHom.mapIntegralClosure [Algebra R S] (f : A →ₐ[R] S) :
     integralClosure R A →ₐ[R] integralClosure R S :=
   (f.restrictDomain (integralClosure R A)).codRestrict (integralClosure R S) (fun ⟨_, h⟩ => h.map f)
 
 @[simp]
-theorem integralClosure_coe_algHom_restrict [Algebra R S] (f : A →ₐ[R] S)
-    (x : integralClosure R A) : (integralClosure_algHom_restrict f x : S) = f (x : A) := rfl
+theorem AlgHom.coe_mapIntegralClosure [Algebra R S] (f : A →ₐ[R] S)
+    (x : integralClosure R A) : (f.mapIntegralClosure x : S) = f (x : A) := rfl
 
 /-- An `AlgEquiv` between two rings restrict to an `AlgEquiv` between the integral closures inside
 them. -/
-noncomputable def integralClosure_algEquiv_restrict [Algebra R S] (f : A ≃ₐ[R] S) :
-    integralClosure R A ≃ₐ[R] integralClosure R S := by
-  refine AlgEquiv.ofBijective (integralClosure_algHom_restrict f) ⟨?_, ?_⟩
-  · erw [AlgHom.injective_codRestrict]
-    exact (EquivLike.injective f).comp Subtype.val_injective
-  · rintro ⟨y, hy⟩
-    exact ⟨⟨f.symm y, hy.map f.symm⟩, by rw [Subtype.mk_eq_mk]; simp⟩
+def AlgEquiv.mapIntegralClosure [Algebra R S] (f : A ≃ₐ[R] S) :
+    integralClosure R A ≃ₐ[R] integralClosure R S :=
+  AlgEquiv.ofAlgHom (f : A →ₐ[R] S).mapIntegralClosure (f.symm : S →ₐ[R] A).mapIntegralClosure
+    (AlgHom.ext fun _ ↦ Subtype.ext (f.right_inv _))
+    (AlgHom.ext fun _ ↦ Subtype.ext (f.left_inv _))
 
 @[simp]
-theorem integralClosure_coe_algEquiv_restrict [Algebra R S] (f : A ≃ₐ[R] S)
-    (x : integralClosure R A) : (integralClosure_algEquiv_restrict f x : S) = f (x : A) := rfl
+theorem AlgEquiv.coe_mapIntegralClosure [Algebra R S] (f : A ≃ₐ[R] S)
+    (x : integralClosure R A) : (f.mapIntegralClosure x : S) = f (x : A) := rfl
 
 theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
   let ⟨p, hpm, hpx⟩ := x.2

To fit with the "please try harder" convention of ! tactics.

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

@@ -267,7 +267,7 @@ theorem isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R
       (by ext; apply one_smul)
       (by intros x y; ext; apply mul_smul)
   obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by
-    by_contra' h'
+    by_contra! h'
     apply hN
     rwa [eq_bot_iff]
   have : Function.Injective f := by

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

@@ -267,8 +267,7 @@ theorem isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R
       (by ext; apply one_smul)
       (by intros x y; ext; apply mul_smul)
   obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by
-    by_contra h'
-    push_neg at h'
+    by_contra' h'
     apply hN
     rwa [eq_bot_iff]
   have : Function.Injective f := by

@@ -468,6 +468,30 @@ theorem integralClosure_map_algEquiv [Algebra R S] (f : A ≃ₐ[R] S) :
     simp
 #align integral_closure_map_alg_equiv integralClosure_map_algEquiv
 
+/-- An `AlgHom` between two rings restrict to an `AlgHom` between the integral closures inside
+them. -/
+def integralClosure_algHom_restrict [Algebra R S] (f : A →ₐ[R] S) :
+    integralClosure R A →ₐ[R] integralClosure R S :=
+  (f.restrictDomain (integralClosure R A)).codRestrict (integralClosure R S) (fun ⟨_, h⟩ => h.map f)
+
+@[simp]
+theorem integralClosure_coe_algHom_restrict [Algebra R S] (f : A →ₐ[R] S)
+    (x : integralClosure R A) : (integralClosure_algHom_restrict f x : S) = f (x : A) := rfl
+
+/-- An `AlgEquiv` between two rings restrict to an `AlgEquiv` between the integral closures inside
+them. -/
+noncomputable def integralClosure_algEquiv_restrict [Algebra R S] (f : A ≃ₐ[R] S) :
+    integralClosure R A ≃ₐ[R] integralClosure R S := by
+  refine AlgEquiv.ofBijective (integralClosure_algHom_restrict f) ⟨?_, ?_⟩
+  · erw [AlgHom.injective_codRestrict]
+    exact (EquivLike.injective f).comp Subtype.val_injective
+  · rintro ⟨y, hy⟩
+    exact ⟨⟨f.symm y, hy.map f.symm⟩, by rw [Subtype.mk_eq_mk]; simp⟩
+
+@[simp]
+theorem integralClosure_coe_algEquiv_restrict [Algebra R S] (f : A ≃ₐ[R] S)
+    (x : integralClosure R A) : (integralClosure_algEquiv_restrict f x : S) = f (x : A) := rfl
+
 theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
   let ⟨p, hpm, hpx⟩ := x.2
   ⟨p, hpm,

Zulip

Initially I just wanted to add more dot notations for IsIntegral and IsAlgebraic (done in #8437); then I noticed near-duplicates Algebra.isIntegral_of_finite [Field R] [Ring A] and RingHom.IsIntegral.of_finite [CommRing R] [CommRing A] so I went on to generalize the latter to cover the former, and generalized everything in the IntegralClosure file to the noncommutative case whenever possible.

In the process I noticed more golfs, which result in this PR. Most notably, isIntegral_of_mem_of_FG is now proven using Cayley-Hamilton and doesn't depend on the Noetherian case isIntegral_of_noetherian; the latter is now proven using the former. In total the golfs makes mathlib 227 lines leaner (+487 -714).

The main changes are in the single file RingTheory/IntegralClosure:

• Change the definition of Algebra.IsIntegral which makes it unfold to IsIntegral rather than RingHom.IsIntegralElem because the former has much more APIs.

• Fix lemma names involving is_integral which are actually about IsIntegralElem: RingHom.is_integral_map → RingHom.isIntegralElem_map RingHom.is_integral_of_mem_closure → RingHom.IsIntegralElem.of_mem_closure RingHom.is_integral_zero/one → RingHom.isIntegralElem_zero/one RingHom.is_integral_add/neg/sub/mul/of_mul_unit → RingHom.IsIntegralElem.add/neg/sub/mul/of_mul_unit

• Add a lemma Algebra.IsIntegral.of_injective.

• Move isIntegral_of_(submodule_)noetherian down and golf them.

• Remove (Algebra.)isIntegral_of_finite that work only over fields, in favor of the more general (Algebra.)isIntegral.of_finite.

• Merge duplicate lemmas isIntegral_of_isScalarTower and isIntegral_tower_top_of_isIntegral into IsIntegral.tower_top.

• Golf IsIntegral.of_mem_of_fg by first proving IsIntegral.of_finite using Cayley-Hamilton.

• Add a docstring mentioning the Kurosh problem at Algebra.IsIntegral.finite. The negative solution to the problem means the theorem doesn't generalize to noncommutative algebras.

• Golf IsIntegral.tmul and isField_of_isIntegral_of_isField(').

• Combine isIntegral_trans_aux into isIntegral_trans and golf.

• Add Algebra namespace to isIntegral_sup.

• rename lemmas for dot notation: RingHom.isIntegral_trans → RingHom.IsIntegral.trans RingHom.isIntegral_quotient/tower_bot/top_of_isIntegral → RingHom.IsIntegral.quotient/tower_bot/top isIntegral_of_mem_closure' → IsIntegral.of_mem_closure' (and the '' version) isIntegral_of_surjective → Algebra.isIntegral_of_surjective

The next changed file is RingTheory/Algebraic:

• Rename: of_larger_base → tower_top (for consistency with IsIntegral) Algebra.isAlgebraic_of_finite → Algebra.IsAlgebraic.of_finite Algebra.isAlgebraic_trans → Algebra.IsAlgebraic.trans

• Add new lemmasAlgebra.IsIntegral.isAlgebraic, isAlgebraic_algHom_iff, and Algebra.IsAlgebraic.of_injective to streamline some proofs.

The generalization from CommRing to Ring requires an additional lemma scaleRoots_eval₂_mul_of_commute in Polynomial/ScaleRoots.

A lemma Algebra.lmul_injective is added to Algebra/Bilinear (in order to golf the proof of IsIntegral.of_mem_of_fg).

In all other files, I merely fix the changed names, or use newly available dot notations.

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

@@ -66,78 +66,26 @@ variable (A)
 
 /-- An algebra is integral if every element of the extension is integral over the base ring -/
 protected def Algebra.IsIntegral : Prop :=
-  (algebraMap R A).IsIntegral
+  ∀ x : A, IsIntegral R x
 #align algebra.is_integral Algebra.IsIntegral
 
 variable {R A}
 
-theorem RingHom.is_integral_map {x : R} : f.IsIntegralElem (f x) :=
+theorem RingHom.isIntegralElem_map {x : R} : f.IsIntegralElem (f x) :=
   ⟨X - C x, monic_X_sub_C _, by simp⟩
-#align ring_hom.is_integral_map RingHom.is_integral_map
+#align ring_hom.is_integral_map RingHom.isIntegralElem_map
 
 theorem isIntegral_algebraMap {x : R} : IsIntegral R (algebraMap R A x) :=
-  (algebraMap R A).is_integral_map
+  (algebraMap R A).isIntegralElem_map
 #align is_integral_algebra_map isIntegral_algebraMap
 
-theorem isIntegral_of_noetherian (H : IsNoetherian R A) (x : A) : IsIntegral R x := by
-  let leval : R[X] →ₗ[R] A := (aeval x).toLinearMap
-  let D : ℕ → Submodule R A := fun n => (degreeLE R n).map leval
-  let M := WellFounded.min (isNoetherian_iff_wellFounded.1 H) (Set.range D) ⟨_, ⟨0, rfl⟩⟩
-  have HM : M ∈ Set.range D := WellFounded.min_mem _ _ _
-  cases' HM with N HN
-  have HM : ¬M < D (N + 1) :=
-    WellFounded.not_lt_min (isNoetherian_iff_wellFounded.1 H) (Set.range D) _ ⟨N + 1, rfl⟩
-  rw [← HN] at HM
-  have HN2 : D (N + 1) ≤ D N :=
-    _root_.by_contradiction fun H =>
-      HM (lt_of_le_not_le (map_mono (degreeLE_mono (WithBot.coe_le_coe.2 (Nat.le_succ N)))) H)
-  have HN3 : leval (X ^ (N + 1)) ∈ D N := HN2 (mem_map_of_mem (mem_degreeLE.2 (degree_X_pow_le _)))
-  rcases HN3 with ⟨p, hdp, hpe⟩
-  refine' ⟨X ^ (N + 1) - p, monic_X_pow_sub (mem_degreeLE.1 hdp), _⟩
-  show leval (X ^ (N + 1) - p) = 0
-  rw [LinearMap.map_sub, hpe, sub_self]
-#align is_integral_of_noetherian isIntegral_of_noetherian
-
-theorem isIntegral_of_submodule_noetherian (S : Subalgebra R A)
-    (H : IsNoetherian R (Subalgebra.toSubmodule S)) (x : A) (hx : x ∈ S) : IsIntegral R x := by
-  suffices IsIntegral R (show S from ⟨x, hx⟩) by
-    rcases this with ⟨p, hpm, hpx⟩
-    replace hpx := congr_arg S.val hpx
-    refine' ⟨p, hpm, Eq.trans _ hpx⟩
-    simp only [aeval_def, eval₂, sum_def]
-    rw [S.val.map_sum]
-    refine' Finset.sum_congr rfl fun n _hn => _
-    rw [S.val.map_mul, S.val.map_pow, S.val.commutes, S.val_apply, Subtype.coe_mk]
-  refine' isIntegral_of_noetherian H ⟨x, hx⟩
-#align is_integral_of_submodule_noetherian isIntegral_of_submodule_noetherian
-
 end Ring
 
 section
 
-variable {K A : Type*}
-
-variable [Field K] [Ring A] [Algebra K A] [FiniteDimensional K A]
-
-variable (K)
-
-theorem IsIntegral.of_finite (e : A) : IsIntegral K e :=
-  isIntegral_of_noetherian (IsNoetherian.iff_fg.2 inferInstance) _
-
-variable (A)
-
-/-- A field extension is integral if it is finite. -/
-theorem Algebra.isIntegral_of_finite : Algebra.IsIntegral K A := fun x =>
-  isIntegral_of_submodule_noetherian ⊤ (IsNoetherian.iff_fg.2 inferInstance) x Algebra.mem_top
-#align algebra.is_integral_of_finite Algebra.isIntegral_of_finite
-
-end
-
-section
-
 variable {R A B S : Type*}
 
-variable [CommRing R] [CommRing A] [CommRing B] [CommRing S]
+variable [CommRing R] [CommRing A] [Ring B] [CommRing S]
 
 variable [Algebra R A] [Algebra R B] (f : R →+* S)
 
@@ -146,54 +94,61 @@ theorem IsIntegral.map {B C F : Type*} [Ring B] [Ring C] [Algebra R B] [Algebra
     (hb : IsIntegral R b) : IsIntegral R (f b) := by
   obtain ⟨P, hP⟩ := hb
   refine' ⟨P, hP.1, _⟩
-  rw [← aeval_def, show (aeval (f b)) P = (aeval (f b)) (P.map (algebraMap R A)) by simp,
+  rw [← aeval_def, ← aeval_map_algebraMap A,
     aeval_algHom_apply, aeval_map_algebraMap, aeval_def, hP.2, _root_.map_zero]
 #align map_is_integral IsIntegral.map
 
-theorem IsIntegral.map_of_comp_eq {R S T U : Type*} [CommRing R] [CommRing S]
-    [CommRing T] [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
+theorem IsIntegral.map_of_comp_eq {R S T U : Type*} [CommRing R] [Ring S]
+    [CommRing T] [Ring U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
     (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
-    IsIntegral T (ψ a) := by
-  rw [IsIntegral, RingHom.IsIntegralElem] at ha ⊢
-  obtain ⟨p, hp⟩ := ha
-  refine' ⟨p.map φ, hp.left.map _, _⟩
-  rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply, eval_map, hp.right,
-    RingHom.map_zero]
+    IsIntegral T (ψ a) :=
+  let ⟨p, hp⟩ := ha
+  ⟨p.map φ, hp.1.map _, by
+    rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply, eval_map, hp.2, ψ.map_zero]⟩
 #align is_integral_map_of_comp_eq_of_is_integral IsIntegral.map_of_comp_eq
 
-theorem isIntegral_algHom_iff {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
-    (f : A →ₐ[R] B) (hf : Function.Injective f) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x := by
-  refine' ⟨_, IsIntegral.map f⟩
-  rintro ⟨p, hp, hx⟩
-  use p, hp
-  rwa [← f.comp_algebraMap, ← AlgHom.coe_toRingHom, ← Polynomial.hom_eval₂, AlgHom.coe_toRingHom,
+section
+
+variable {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
+variable (f : A →ₐ[R] B) (hf : Function.Injective f)
+
+theorem isIntegral_algHom_iff {x : A} : IsIntegral R (f x) ↔ IsIntegral R x := by
+  refine ⟨fun ⟨p, hp, hx⟩ ↦ ⟨p, hp, ?_⟩, IsIntegral.map f⟩
+  rwa [← f.comp_algebraMap, ← AlgHom.coe_toRingHom, ← hom_eval₂, AlgHom.coe_toRingHom,
     map_eq_zero_iff f hf] at hx
 #align is_integral_alg_hom_iff isIntegral_algHom_iff
 
+theorem Algebra.IsIntegral.of_injective (h : Algebra.IsIntegral R B) : Algebra.IsIntegral R A :=
+  fun _ ↦ (isIntegral_algHom_iff f hf).mp (h _)
+
+end
+
 @[simp]
 theorem isIntegral_algEquiv {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
-  ⟨fun h => by simpa using IsIntegral.map f.symm.toAlgHom h, IsIntegral.map f.toAlgHom⟩
+  ⟨fun h ↦ by simpa using h.map f.symm, IsIntegral.map f⟩
 #align is_integral_alg_equiv isIntegral_algEquiv
 
-theorem isIntegral_of_isScalarTower [Algebra A B] [IsScalarTower R A B] {x : B}
+/-- If `R → A → B` is an algebra tower,
+then if the entire tower is an integral extension so is `A → B`. -/
+theorem IsIntegral.tower_top [Algebra A B] [IsScalarTower R A B] {x : B}
     (hx : IsIntegral R x) : IsIntegral A x :=
   let ⟨p, hp, hpx⟩ := hx
   ⟨p.map <| algebraMap R A, hp.map _, by rw [← aeval_def, aeval_map_algebraMap, aeval_def, hpx]⟩
-#align is_integral_of_is_scalar_tower isIntegral_of_isScalarTower
+#align is_integral_of_is_scalar_tower IsIntegral.tower_top
+#align is_integral_tower_top_of_is_integral IsIntegral.tower_top
 
 theorem map_isIntegral_int {B C F : Type*} [Ring B] [Ring C] {b : B} [RingHomClass F B C] (f : F)
     (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
-  IsIntegral.map (f : B →+* C).toIntAlgHom hb
+  hb.map (f : B →+* C).toIntAlgHom
 #align map_is_integral_int map_isIntegral_int
 
-theorem IsIntegral.of_subring {x : A} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
-  isIntegral_of_isScalarTower hx
+theorem IsIntegral.of_subring {x : B} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
+  hx.tower_top
 #align is_integral_of_subring IsIntegral.of_subring
 
 protected theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A}
-    (h : IsIntegral R x) :
-    IsIntegral R (algebraMap A B x) := by
+    (h : IsIntegral R x) : IsIntegral R (algebraMap A B x) := by
   rcases h with ⟨f, hf, hx⟩
   use f, hf
   rw [IsScalarTower.algebraMap_eq R A B, ← hom_eval₂, hx, RingHom.map_zero]
@@ -205,44 +160,31 @@ theorem isIntegral_algebraMap_iff [Algebra A B] [IsScalarTower R A B] {x : A}
   isIntegral_algHom_iff (IsScalarTower.toAlgHom R A B) hAB
 #align is_integral_algebra_map_iff isIntegral_algebraMap_iff
 
-theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
+theorem isIntegral_iff_isIntegral_closure_finite {r : B} :
     IsIntegral R r ↔ ∃ s : Set R, s.Finite ∧ IsIntegral (Subring.closure s) r := by
   constructor <;> intro hr
   · rcases hr with ⟨p, hmp, hpr⟩
     refine' ⟨_, Finset.finite_toSet _, p.restriction, monic_restriction.2 hmp, _⟩
     rw [← aeval_def, ← aeval_map_algebraMap R r p.restriction, map_restriction, aeval_def, hpr]
   rcases hr with ⟨s, _, hsr⟩
-  exact IsIntegral.of_subring _ hsr
+  exact hsr.of_subring _
 #align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finite
 
-theorem IsIntegral.fg_adjoin_singleton (x : A) (hx : IsIntegral R x) :
-    (Algebra.adjoin R ({x} : Set A)).toSubmodule.FG := by
+theorem IsIntegral.fg_adjoin_singleton {x : B} (hx : IsIntegral R x) :
+    (Algebra.adjoin R {x}).toSubmodule.FG := by
   rcases hx with ⟨f, hfm, hfx⟩
-  exists Finset.image ((· ^ ·) x) (Finset.range (natDegree f + 1))
-  apply le_antisymm
-  · rw [span_le]
-    intro s hs
-    rw [Finset.mem_coe] at hs
-    rcases Finset.mem_image.1 hs with ⟨k, hk, rfl⟩
-    clear hk
-    exact (Algebra.adjoin R {x}).pow_mem (Algebra.subset_adjoin (Set.mem_singleton _)) k
-  intro r hr; change r ∈ Algebra.adjoin R ({x} : Set A) at hr
-  rw [Algebra.adjoin_singleton_eq_range_aeval] at hr
+  use (Finset.range <| f.natDegree + 1).image (x ^ ·)
+  refine (span_le.mpr fun s hs ↦ ?_).antisymm fun r hr ↦ ?_
+  · rcases Finset.mem_image.1 hs with ⟨k, -, rfl⟩
+    exact (Algebra.adjoin R {x}).pow_mem (Algebra.subset_adjoin rfl) k
+  rw [Subalgebra.mem_toSubmodule, Algebra.adjoin_singleton_eq_range_aeval] at hr
   rcases (aeval x).mem_range.mp hr with ⟨p, rfl⟩
-  rw [← modByMonic_add_div p hfm]
-  rw [← aeval_def] at hfx
-  rw [AlgHom.map_add, AlgHom.map_mul, hfx, zero_mul, add_zero]
-  have : degree (p %ₘ f) ≤ degree f := degree_modByMonic_le p hfm
-  generalize p %ₘ f = q at this ⊢
-  rw [← sum_C_mul_X_pow_eq q, aeval_def, eval₂_sum, sum_def]
-  refine' sum_mem fun k hkq => _
-  rw [eval₂_mul, eval₂_C, eval₂_pow, eval₂_X, ← Algebra.smul_def]
-  refine' smul_mem _ _ (subset_span _)
-  rw [Finset.mem_coe]; refine' Finset.mem_image.2 ⟨_, _, rfl⟩
-  rw [Finset.mem_range, Nat.lt_succ_iff]; refine' le_of_not_lt fun hk => _
-  rw [degree_le_iff_coeff_zero] at this
-  rw [mem_support_iff] at hkq; apply hkq; apply this
-  exact lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 hk)
+  rw [← modByMonic_add_div p hfm, map_add, map_mul, aeval_def x f, hfx,
+      zero_mul, add_zero, ← sum_C_mul_X_pow_eq (p %ₘ f), aeval_def, eval₂_sum, sum_def]
+  refine sum_mem fun k hkq ↦ ?_
+  rw [C_mul_X_pow_eq_monomial, eval₂_monomial, ← Algebra.smul_def]
+  exact smul_mem _ _ (subset_span <| Finset.mem_image_of_mem _ <| Finset.mem_range_succ_iff.mpr <|
+    (le_natDegree_of_mem_supp _ hkq).trans <| natDegree_modByMonic_le p hfm)
 #align fg_adjoin_singleton_of_integral IsIntegral.fg_adjoin_singleton
 
 theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
@@ -256,123 +198,48 @@ theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
       rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule]
       exact
         FG.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)
-          (IsIntegral.fg_adjoin_singleton _ <| his a <| Set.mem_insert a s))
+          (his a <| Set.mem_insert a s).fg_adjoin_singleton)
     his
 #align fg_adjoin_of_finite fg_adjoin_of_finite
 
 theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
-    (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (↑s : Set A)) :=
+    (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (s : Set A)) :=
   isNoetherian_of_fg_of_noetherian _ (fg_adjoin_of_finite s.finite_toSet hs)
 #align is_noetherian_adjoin_finset isNoetherian_adjoin_finset
 
-/-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
-  then all elements of `S` are integral over `R`. -/
-theorem IsIntegral.of_mem_of_fg (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) :
-    IsIntegral R x := by
-  -- say `x ∈ S`. We want to prove that `x` is integral over `R`.
-  -- Say `S` is generated as an `R`-module by the set `y`.
-  cases' HS with y hy
-  -- We can write `x` as `∑ rᵢ yᵢ` for `yᵢ ∈ Y`.
-  obtain ⟨lx, hlx1, hlx2⟩ :
-    ∃ (l : A →₀ R), l ∈ Finsupp.supported R R ↑y ∧ (Finsupp.total A A R id) l = x := by
-    rwa [← @Finsupp.mem_span_image_iff_total A A R _ _ _ id (↑y) x, Set.image_id (y : Set A), hy]
-  -- Note that `y ⊆ S`.
-  have hyS : ∀ {p}, p ∈ y → p ∈ S := fun {p} hp =>
-    show p ∈ Subalgebra.toSubmodule S by
-      rw [← hy]
-      exact subset_span hp
-  -- Now `S` is a subalgebra so the product of two elements of `y` is also in `S`.
-  have : ∀ jk : (y ×ˢ y : Finset (A × A)),
-      jk.1.1 * jk.1.2 ∈ (Subalgebra.toSubmodule S) := fun jk =>
-    S.mul_mem (hyS (Finset.mem_product.1 jk.2).1) (hyS (Finset.mem_product.1 jk.2).2)
-  rw [← hy, ← Set.image_id (y : Set A)] at this
-  simp only [Finsupp.mem_span_image_iff_total] at this
-  -- Say `yᵢyⱼ = ∑rᵢⱼₖ yₖ`
-  choose ly hly1 hly2 using this
-  -- Now let `S₀` be the subring of `R` generated by the `rᵢ` and the `rᵢⱼₖ`.
-  let S₀ : Subring R :=
-    Subring.closure ↑(lx.frange ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))
-  -- It suffices to prove that `x` is integral over `S₀`.
-  refine' IsIntegral.of_subring S₀ _
-  letI : CommRing S₀ := SubringClass.toCommRing S₀
-  letI : Algebra S₀ A := Algebra.ofSubring S₀
-  -- Claim: the `S₀`-module span (in `A`) of the set `y ∪ {1}` is closed under
-  -- multiplication (indeed, this is the motivation for the definition of `S₀`).
-  have :
-    span S₀ (insert 1 ↑y : Set A) * span S₀ (insert 1 ↑y : Set A) ≤
-      span S₀ (insert 1 ↑y : Set A) := by
-    rw [span_mul_span]
-    refine' span_le.2 fun z hz => _
-    rcases Set.mem_mul.1 hz with ⟨p, q, rfl | hp, hq, rfl⟩
-    · rw [one_mul]
-      exact subset_span hq
-    rcases hq with (rfl | hq)
-    · rw [mul_one]
-      exact subset_span (Or.inr hp)
-    erw [← hly2 ⟨(p, q), Finset.mem_product.2 ⟨hp, hq⟩⟩]
-    rw [Finsupp.total_apply, Finsupp.sum]
-    refine' (span S₀ (insert 1 ↑y : Set A)).sum_mem fun t ht => _
-    have : ly ⟨(p, q), Finset.mem_product.2 ⟨hp, hq⟩⟩ t ∈ S₀ :=
-      Subring.subset_closure
-        (Finset.mem_union_right _ <|
-          Finset.mem_biUnion.2
-            ⟨⟨(p, q), Finset.mem_product.2 ⟨hp, hq⟩⟩, Finset.mem_univ _,
-              Finsupp.mem_frange.2 ⟨Finsupp.mem_support_iff.1 ht, _, rfl⟩⟩)
-    change (⟨_, this⟩ : S₀) • t ∈ _
-    exact smul_mem _ _ (subset_span <| Or.inr <| hly1 _ ht)
-  -- Hence this span is a subring. Call this subring `S₁`.
-  let S₁ : Subring A :=
-    { carrier := span S₀ (insert 1 ↑y : Set A)
-      one_mem' := subset_span <| Or.inl rfl
-      mul_mem' := fun {p q} hp hq => this <| mul_mem_mul hp hq
-      zero_mem' := (span S₀ (insert 1 ↑y : Set A)).zero_mem
-      add_mem' := fun {_ _} => (span S₀ (insert 1 ↑y : Set A)).add_mem
-      neg_mem' := fun {_} => (span S₀ (insert 1 ↑y : Set A)).neg_mem }
-  have : S₁ = Subalgebra.toSubring (Algebra.adjoin S₀ (↑y : Set A)) := by
-    ext z
-    suffices
-      z ∈ span (↥S₀) (insert 1 ↑y : Set A) ↔
-        z ∈ Subalgebra.toSubmodule (Algebra.adjoin (↥S₀) (y : Set A)) by
-      simpa
-    constructor <;> intro hz
-    · exact
-        (span_le.2
-          (Set.insert_subset_iff.2
-            ⟨(Algebra.adjoin S₀ (y : Set A)).one_mem, Algebra.subset_adjoin⟩)) hz
-    · rw [Subalgebra.mem_toSubmodule, Algebra.mem_adjoin_iff] at hz
-      suffices Subring.closure (Set.range (algebraMap (↥S₀) A) ∪ ↑y) ≤ S₁ by exact this hz
-      refine' Subring.closure_le.2 (Set.union_subset _ fun t ht => subset_span <| Or.inr ht)
-      rw [Set.range_subset_iff]
-      intro y'
-      rw [Algebra.algebraMap_eq_smul_one]
-      exact smul_mem (span S₀ (insert (1 : A) (y : Set A))) y' (subset_span (Or.inl rfl))
-  have foo : ∀ z, z ∈ S₁ ↔ z ∈ Algebra.adjoin (↥S₀) (y : Set A)
-  simp only [this, Finset.univ_eq_attach, Subalgebra.mem_toSubring, forall_const]
-  haveI : IsNoetherianRing S₀ := is_noetherian_subring_closure _ (Finset.finite_toSet _)
-  refine'
-    isIntegral_of_submodule_noetherian (Algebra.adjoin S₀ ↑y)
-      (isNoetherian_of_fg_of_noetherian _
-        ⟨insert 1 y, by
-          rw [Finset.coe_insert]
-          ext z
-          simp only [Finset.coe_sort_coe, Finset.univ_eq_attach, Finset.mem_coe,
-            Subalgebra.mem_toSubmodule]
-          convert foo z⟩)
-      _ _
-  rw [← hlx2, Finsupp.total_apply, Finsupp.sum]
-  refine' Subalgebra.sum_mem _ fun r hr => _
-  have : lx r ∈ S₀ :=
-    Subring.subset_closure (Finset.mem_union_left _ (Finset.mem_image_of_mem _ hr))
-  change (⟨_, this⟩ : S₀) • r ∈ _
-  rw [Finsupp.mem_supported] at hlx1
-  exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _
-#align is_integral_of_mem_of_fg IsIntegral.of_mem_of_fg
-
 theorem Module.End.isIntegral {M : Type*} [AddCommGroup M] [Module R M] [Module.Finite R M] :
     Algebra.IsIntegral R (Module.End R M) :=
   LinearMap.exists_monic_and_aeval_eq_zero R
 #align module.End.is_integral Module.End.isIntegral
 
+variable (R)
+theorem IsIntegral.of_finite [Module.Finite R B] (x : B) : IsIntegral R x :=
+  (isIntegral_algHom_iff (Algebra.lmul R B) Algebra.lmul_injective).mp (Module.End.isIntegral _)
+
+variable (B)
+theorem Algebra.IsIntegral.of_finite [Module.Finite R B] : Algebra.IsIntegral R B :=
+  .of_finite R
+#align algebra.is_integral.of_finite Algebra.IsIntegral.of_finite
+
+variable {R B}
+
+/-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
+  then all elements of `S` are integral over `R`. -/
+theorem IsIntegral.of_mem_of_fg {A} [Ring A] [Algebra R A] (S : Subalgebra R A)
+    (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) : IsIntegral R x :=
+  have : Module.Finite R S := ⟨(fg_top _).mpr HS⟩
+  (isIntegral_algHom_iff S.val Subtype.val_injective).mpr (.of_finite R (⟨x, hx⟩ : S))
+#align is_integral_of_mem_of_fg IsIntegral.of_mem_of_fg
+
+theorem isIntegral_of_noetherian (_ : IsNoetherian R B) (x : B) : IsIntegral R x :=
+  .of_finite R x
+#align is_integral_of_noetherian isIntegral_of_noetherian
+
+theorem isIntegral_of_submodule_noetherian (S : Subalgebra R B)
+    (H : IsNoetherian R (Subalgebra.toSubmodule S)) (x : B) (hx : x ∈ S) : IsIntegral R x :=
+  .of_mem_of_fg _ ((fg_top _).mp <| H.noetherian _) _ hx
+#align is_integral_of_submodule_noetherian isIntegral_of_submodule_noetherian
+
 /-- Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
 and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. -/
 theorem isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R M] [Module A M]
@@ -420,19 +287,31 @@ variable {f}
 
 theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
   letI := f.toAlgebra
-  fun _ => IsIntegral.of_mem_of_fg ⊤ h.1 _ trivial
+  fun _ ↦ IsIntegral.of_mem_of_fg ⊤ h.1 _ trivial
 #align ring_hom.finite.to_is_integral RingHom.Finite.to_isIntegral
 
 alias RingHom.IsIntegral.of_finite := RingHom.Finite.to_isIntegral
 #align ring_hom.is_integral.of_finite RingHom.IsIntegral.of_finite
 
-theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite := by
+/-- The [Kurosh problem](https://en.wikipedia.org/wiki/Kurosh_problem) asks to show that
+  this is still true when `A` is not necessarily commutative and `R` is a field, but it has
+  been solved in the negative. See https://arxiv.org/pdf/1706.02383.pdf for criteria for a
+  finitely generated algebraic (= integral) algebra over a field to be finite dimensional. -/
+theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.FiniteType R A] :
+    Module.Finite R A :=
+  have ⟨s, hs⟩ := h'
+  ⟨by apply hs ▸ fg_adjoin_of_finite s.finite_toSet fun x _ ↦ h x⟩
+#align algebra.is_integral.finite Algebra.IsIntegral.finite
+
+/-- finite = integral + finite type -/
+theorem Algebra.finite_iff_isIntegral_and_finiteType :
+    Module.Finite R A ↔ Algebra.IsIntegral R A ∧ Algebra.FiniteType R A :=
+  ⟨fun _ ↦ ⟨.of_finite R, inferInstance⟩, fun ⟨h, _⟩ ↦ h.finite⟩
+#align algebra.finite_iff_is_integral_and_finite_type Algebra.finite_iff_isIntegral_and_finiteType
+
+theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite :=
   letI := f.toAlgebra
-  obtain ⟨s, hs⟩ := h'
-  constructor
-  change (⊤ : Subalgebra R S).toSubmodule.FG
-  rw [← hs]
-  exact fg_adjoin_of_finite (Set.toFinite _) fun x _ => h x
+  Algebra.IsIntegral.finite h (h' := h')
 #align ring_hom.is_integral.to_finite RingHom.IsIntegral.to_finite
 
 alias RingHom.Finite.of_isIntegral_of_finiteType := RingHom.IsIntegral.to_finite
@@ -440,120 +319,98 @@ alias RingHom.Finite.of_isIntegral_of_finiteType := RingHom.IsIntegral.to_finite
 
 /-- finite = integral + finite type -/
 theorem RingHom.finite_iff_isIntegral_and_finiteType : f.Finite ↔ f.IsIntegral ∧ f.FiniteType :=
-  ⟨fun h => ⟨h.to_isIntegral, h.to_finiteType⟩, fun ⟨h, h'⟩ => h.to_finite h'⟩
+  ⟨fun h ↦ ⟨h.to_isIntegral, h.to_finiteType⟩, fun ⟨h, h'⟩ ↦ h.to_finite h'⟩
 #align ring_hom.finite_iff_is_integral_and_finite_type RingHom.finite_iff_isIntegral_and_finiteType
 
-theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.FiniteType R A] :
-    Module.Finite R A := by
-  have :=
-    h.to_finite
-      (by
-        rw [RingHom.FiniteType]
-        convert h'
-        -- Porting note: was `ext`
-        refine IsScalarTower.Algebra.ext (algebraMap R A).toAlgebra _ fun r x => ?_
-        exact (Algebra.smul_def _ _).symm)
-  rw [RingHom.Finite] at this
-  convert this
-  ext
-  exact Algebra.smul_def _ _
-#align algebra.is_integral.finite Algebra.IsIntegral.finite
-
-theorem Algebra.IsIntegral.of_finite [h : Module.Finite R A] : Algebra.IsIntegral R A :=
-  fun _ ↦ IsIntegral.of_mem_of_fg ⊤ h.1 _ trivial
-#align algebra.is_integral.of_finite Algebra.IsIntegral.of_finite
-
-/-- finite = integral + finite type -/
-theorem Algebra.finite_iff_isIntegral_and_finiteType :
-    Module.Finite R A ↔ Algebra.IsIntegral R A ∧ Algebra.FiniteType R A :=
-  ⟨fun _ => ⟨Algebra.IsIntegral.of_finite, inferInstance⟩, fun ⟨h, _⟩ => h.finite⟩
-#align algebra.finite_iff_is_integral_and_finite_type Algebra.finite_iff_isIntegral_and_finiteType
-
 variable (f)
 
-theorem RingHom.is_integral_of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
+theorem RingHom.IsIntegralElem.of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
     (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z := by
   letI : Algebra R S := f.toAlgebra
-  have := (IsIntegral.fg_adjoin_singleton x hx).mul (IsIntegral.fg_adjoin_singleton y hy)
+  have := (IsIntegral.fg_adjoin_singleton hx).mul (IsIntegral.fg_adjoin_singleton hy)
   rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this
   exact
     IsIntegral.of_mem_of_fg (Algebra.adjoin R {x, y}) this z
       (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)
-#align ring_hom.is_integral_of_mem_closure RingHom.is_integral_of_mem_closure
+#align ring_hom.is_integral_of_mem_closure RingHom.IsIntegralElem.of_mem_closure
 
-theorem IsIntegral.of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
+nonrec theorem IsIntegral.of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
     (hz : z ∈ Subring.closure ({x, y} : Set A)) : IsIntegral R z :=
-  (algebraMap R A).is_integral_of_mem_closure hx hy hz
+  hx.of_mem_closure (algebraMap R A) hy hz
 #align is_integral_of_mem_closure IsIntegral.of_mem_closure
 
-theorem RingHom.is_integral_zero : f.IsIntegralElem 0 :=
-  f.map_zero ▸ f.is_integral_map
-#align ring_hom.is_integral_zero RingHom.is_integral_zero
+variable (f : R →+* B)
+
+theorem RingHom.isIntegralElem_zero : f.IsIntegralElem 0 :=
+  f.map_zero ▸ f.isIntegralElem_map
+#align ring_hom.is_integral_zero RingHom.isIntegralElem_zero
 
-theorem isIntegral_zero : IsIntegral R (0 : A) :=
-  (algebraMap R A).is_integral_zero
+theorem isIntegral_zero : IsIntegral R (0 : B) :=
+  (algebraMap R B).isIntegralElem_zero
 #align is_integral_zero isIntegral_zero
 
-theorem RingHom.is_integral_one : f.IsIntegralElem 1 :=
-  f.map_one ▸ f.is_integral_map
-#align ring_hom.is_integral_one RingHom.is_integral_one
+theorem RingHom.isIntegralElem_one : f.IsIntegralElem 1 :=
+  f.map_one ▸ f.isIntegralElem_map
+#align ring_hom.is_integral_one RingHom.isIntegralElem_one
 
-theorem isIntegral_one : IsIntegral R (1 : A) :=
-  (algebraMap R A).is_integral_one
+theorem isIntegral_one : IsIntegral R (1 : B) :=
+  (algebraMap R B).isIntegralElem_one
 #align is_integral_one isIntegral_one
 
-theorem RingHom.is_integral_add {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
+theorem RingHom.IsIntegralElem.add (f : R →+* S) {x y : S}
+    (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x + y) :=
-  f.is_integral_of_mem_closure hx hy <|
+  hx.of_mem_closure f hy <|
     Subring.add_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl))
-#align ring_hom.is_integral_add RingHom.is_integral_add
+#align ring_hom.is_integral_add RingHom.IsIntegralElem.add
 
-theorem IsIntegral.add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
+nonrec theorem IsIntegral.add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x + y) :=
-  (algebraMap R A).is_integral_add hx hy
+  hx.add (algebraMap R A) hy
 #align is_integral_add IsIntegral.add
 
-theorem RingHom.is_integral_neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) :=
-  f.is_integral_of_mem_closure hx hx (Subring.neg_mem _ (Subring.subset_closure (Or.inl rfl)))
-#align ring_hom.is_integral_neg RingHom.is_integral_neg
+variable (f : R →+* S)
 
-theorem IsIntegral.neg {x : A} (hx : IsIntegral R x) : IsIntegral R (-x) :=
-  (algebraMap R A).is_integral_neg hx
+-- can be generalized to noncommutative S.
+theorem RingHom.IsIntegralElem.neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) :=
+  hx.of_mem_closure f hx (Subring.neg_mem _ (Subring.subset_closure (Or.inl rfl)))
+#align ring_hom.is_integral_neg RingHom.IsIntegralElem.neg
+
+theorem IsIntegral.neg {x : B} (hx : IsIntegral R x) : IsIntegral R (-x) :=
+  .of_mem_of_fg _ hx.fg_adjoin_singleton _ (Subalgebra.neg_mem _ <| Algebra.subset_adjoin rfl)
 #align is_integral_neg IsIntegral.neg
 
-theorem RingHom.is_integral_sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
+theorem RingHom.IsIntegralElem.sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x - y) := by
-  simpa only [sub_eq_add_neg] using f.is_integral_add hx (f.is_integral_neg hy)
-#align ring_hom.is_integral_sub RingHom.is_integral_sub
+  simpa only [sub_eq_add_neg] using hx.add f (hy.neg f)
+#align ring_hom.is_integral_sub RingHom.IsIntegralElem.sub
 
-theorem IsIntegral.sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
+nonrec theorem IsIntegral.sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x - y) :=
-  (algebraMap R A).is_integral_sub hx hy
+  hx.sub (algebraMap R A) hy
 #align is_integral_sub IsIntegral.sub
 
-theorem RingHom.is_integral_mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
+theorem RingHom.IsIntegralElem.mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x * y) :=
-  f.is_integral_of_mem_closure hx hy
+  hx.of_mem_closure f hy
     (Subring.mul_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl)))
-#align ring_hom.is_integral_mul RingHom.is_integral_mul
+#align ring_hom.is_integral_mul RingHom.IsIntegralElem.mul
 
-theorem IsIntegral.mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
+nonrec theorem IsIntegral.mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x * y) :=
-  (algebraMap R A).is_integral_mul hx hy
+  hx.mul (algebraMap R A) hy
 #align is_integral_mul IsIntegral.mul
 
-theorem IsIntegral.smul [Algebra S A] [Algebra R S] [IsScalarTower R S A] {x : A} (r : R)
-    (hx : IsIntegral S x) : IsIntegral S (r • x) := by
-  rw [Algebra.smul_def, IsScalarTower.algebraMap_apply R S A]
-  exact IsIntegral.mul isIntegral_algebraMap hx
+theorem IsIntegral.smul {R} [CommSemiring R] [CommRing S] [Algebra R B] [Algebra S B] [Algebra R S]
+    [IsScalarTower R S B] {x : B} (r : R)(hx : IsIntegral S x) : IsIntegral S (r • x) :=
+  .of_mem_of_fg _ hx.fg_adjoin_singleton _ <| by
+    rw [← algebraMap_smul S]; apply Subalgebra.smul_mem; exact Algebra.subset_adjoin rfl
 #align is_integral_smul IsIntegral.smul
 
-theorem IsIntegral.of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
+theorem IsIntegral.of_pow {x : B} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
     IsIntegral R x := by
-  rcases hx with ⟨p, ⟨hmonic, heval⟩⟩
-  exact
-    ⟨expand R n p, Monic.expand hn hmonic, by
-      rwa [eval₂_eq_eval_map, map_expand, expand_eval, ← eval₂_eq_eval_map]⟩
+  rcases hx with ⟨p, hmonic, heval⟩
+  exact ⟨expand R n p, hmonic.expand hn, by rwa [← aeval_def, expand_aeval]⟩
 #align is_integral_of_pow IsIntegral.of_pow
 
 variable (R A)
@@ -571,11 +428,11 @@ def integralClosure : Subalgebra R A where
 theorem mem_integralClosure_iff_mem_fg {r : A} :
     r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M :=
   ⟨fun hr =>
-    ⟨Algebra.adjoin R {r}, IsIntegral.fg_adjoin_singleton _ hr, Algebra.subset_adjoin rfl⟩,
-    fun ⟨M, Hf, hrM⟩ => IsIntegral.of_mem_of_fg M Hf _ hrM⟩
+    ⟨Algebra.adjoin R {r}, hr.fg_adjoin_singleton, Algebra.subset_adjoin rfl⟩,
+    fun ⟨M, Hf, hrM⟩ => .of_mem_of_fg M Hf _ hrM⟩
 #align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_fg
 
-variable {R} {A}
+variable {R A}
 
 theorem adjoin_le_integralClosure {x : A} (hx : IsIntegral R x) :
     Algebra.adjoin R {x} ≤ integralClosure R A := by
@@ -592,22 +449,22 @@ theorem le_integralClosure_iff_isIntegral {S : Subalgebra R A} :
         isIntegral_algebraMap_iff Subtype.coe_injective)
 #align le_integral_closure_iff_is_integral le_integralClosure_iff_isIntegral
 
-theorem isIntegral_sup {S T : Subalgebra R A} :
+theorem Algebra.isIntegral_sup {S T : Subalgebra R A} :
     Algebra.IsIntegral R (S ⊔ T : Subalgebra R A) ↔
       Algebra.IsIntegral R S ∧ Algebra.IsIntegral R T := by
   simp only [← le_integralClosure_iff_isIntegral, sup_le_iff]
-#align is_integral_sup isIntegral_sup
+#align is_integral_sup Algebra.isIntegral_sup
 
 /-- Mapping an integral closure along an `AlgEquiv` gives the integral closure. -/
-theorem integralClosure_map_algEquiv (f : A ≃ₐ[R] B) :
-    (integralClosure R A).map (f : A →ₐ[R] B) = integralClosure R B := by
+theorem integralClosure_map_algEquiv [Algebra R S] (f : A ≃ₐ[R] S) :
+    (integralClosure R A).map (f : A →ₐ[R] S) = integralClosure R S := by
   ext y
   rw [Subalgebra.mem_map]
   constructor
   · rintro ⟨x, hx, rfl⟩
-    exact IsIntegral.map f hx
+    exact hx.map f
   · intro hy
-    use f.symm y, IsIntegral.map (f.symm : B →ₐ[R] A) hy
+    use f.symm y, hy.map (f.symm : S →ₐ[R] A)
     simp
 #align integral_closure_map_alg_equiv integralClosure_map_algEquiv
 
@@ -618,41 +475,44 @@ theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
       rwa [← aeval_def, ← Subalgebra.val_apply, aeval_algHom_apply] at hpx⟩
 #align integral_closure.is_integral integralClosure.isIntegral
 
-theorem RingHom.isIntegral_of_isIntegral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
-    (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x := by
-  obtain ⟨p, ⟨p_monic, hp⟩⟩ := hx
-  refine' ⟨scaleRoots p r, ⟨(monic_scaleRoots_iff r).2 p_monic, _⟩⟩
-  convert scaleRoots_eval₂_eq_zero f hp
-  rw [mul_comm x y, ← mul_assoc, hr, one_mul]
-#align ring_hom.is_integral_of_is_integral_mul_unit RingHom.isIntegral_of_isIntegral_mul_unit
-
-theorem IsIntegral.of_mul_unit {x y : A} {r : R} (hr : algebraMap R A r * y = 1)
-    (hx : IsIntegral R (x * y)) : IsIntegral R x :=
-  (algebraMap R A).isIntegral_of_isIntegral_mul_unit x y r hr hx
+theorem IsIntegral.of_mul_unit {x y : B} {r : R} (hr : algebraMap R B r * y = 1)
+    (hx : IsIntegral R (x * y)) : IsIntegral R x := by
+  obtain ⟨p, p_monic, hp⟩ := hx
+  refine ⟨scaleRoots p r, (monic_scaleRoots_iff r).2 p_monic, ?_⟩
+  convert scaleRoots_aeval_eq_zero hp
+  rw [Algebra.commutes] at hr ⊢
+  rw [mul_assoc, hr, mul_one]; rfl
 #align is_integral_of_is_integral_mul_unit IsIntegral.of_mul_unit
 
-/-- Generalization of `isIntegral_of_mem_closure` bootstrapped up from that lemma -/
-theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) :
-    ∀ x ∈ Subring.closure G, IsIntegral R x := fun _ hx =>
-  Subring.closure_induction hx hG isIntegral_zero isIntegral_one (fun _ _ => IsIntegral.add)
-    (fun _ => IsIntegral.neg) fun _ _ => IsIntegral.mul
-#align is_integral_of_mem_closure' isIntegral_of_mem_closure'
+theorem RingHom.IsIntegralElem.of_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
+    (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x :=
+  letI : Algebra R S := f.toAlgebra
+  IsIntegral.of_mul_unit hr hx
+#align ring_hom.is_integral_of_is_integral_mul_unit RingHom.IsIntegralElem.of_mul_unit
+
+/-- Generalization of `IsIntegral.of_mem_closure` bootstrapped up from that lemma -/
+theorem IsIntegral.of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) :
+    ∀ x ∈ Subring.closure G, IsIntegral R x := fun _ hx ↦
+  Subring.closure_induction hx hG isIntegral_zero isIntegral_one (fun _ _ ↦ IsIntegral.add)
+    (fun _ ↦ IsIntegral.neg) fun _ _ ↦ IsIntegral.mul
+#align is_integral_of_mem_closure' IsIntegral.of_mem_closure'
 
-theorem isIntegral_of_mem_closure'' {S : Type*} [CommRing S] {f : R →+* S} (G : Set S)
+theorem IsIntegral.of_mem_closure'' {S : Type*} [CommRing S] {f : R →+* S} (G : Set S)
     (hG : ∀ x ∈ G, f.IsIntegralElem x) : ∀ x ∈ Subring.closure G, f.IsIntegralElem x := fun x hx =>
-  @isIntegral_of_mem_closure' R S _ _ f.toAlgebra G hG x hx
-#align is_integral_of_mem_closure'' isIntegral_of_mem_closure''
+  @IsIntegral.of_mem_closure' R S _ _ f.toAlgebra G hG x hx
+#align is_integral_of_mem_closure'' IsIntegral.of_mem_closure''
 
-theorem IsIntegral.pow {x : A} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (x ^ n) :=
-  (integralClosure R A).pow_mem h n
+theorem IsIntegral.pow {x : B} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (x ^ n) :=
+  .of_mem_of_fg _ h.fg_adjoin_singleton _ <|
+    Subalgebra.pow_mem _ (by exact Algebra.subset_adjoin rfl) _
 #align is_integral.pow IsIntegral.pow
 
-theorem IsIntegral.nsmul {x : A} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (n • x) :=
-  (integralClosure R A).nsmul_mem h n
+theorem IsIntegral.nsmul {x : B} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (n • x) :=
+  h.smul n
 #align is_integral.nsmul IsIntegral.nsmul
 
-theorem IsIntegral.zsmul {x : A} (h : IsIntegral R x) (n : ℤ) : IsIntegral R (n • x) :=
-  (integralClosure R A).zsmul_mem h n
+theorem IsIntegral.zsmul {x : B} (h : IsIntegral R x) (n : ℤ) : IsIntegral R (n • x) :=
+  h.smul n
 #align is_integral.zsmul IsIntegral.zsmul
 
 theorem IsIntegral.multiset_prod {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R x) :
@@ -678,36 +538,21 @@ theorem IsIntegral.sum {α : Type*} {s : Finset α} (f : α → A) (h : ∀ x 
 theorem IsIntegral.det {n : Type*} [Fintype n] [DecidableEq n] {M : Matrix n n A}
     (h : ∀ i j, IsIntegral R (M i j)) : IsIntegral R M.det := by
   rw [Matrix.det_apply]
-  exact IsIntegral.sum _ fun σ _hσ => IsIntegral.zsmul (IsIntegral.prod _ fun i _hi => h _ _) _
+  exact IsIntegral.sum _ fun σ _hσ ↦ (IsIntegral.prod _ fun i _hi => h _ _).zsmul _
 #align is_integral.det IsIntegral.det
 
 @[simp]
 theorem IsIntegral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n) ↔ IsIntegral R x :=
-  ⟨IsIntegral.of_pow hn, fun hx => IsIntegral.pow hx n⟩
+  ⟨IsIntegral.of_pow hn, fun hx ↦ hx.pow n⟩
 #align is_integral.pow_iff IsIntegral.pow_iff
 
 open TensorProduct
 
--- porting note: I fought a lot with this proof.  I tried to follow the original,
--- but was not really able to.
 theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x ⊗ₜ[R] y) := by
-  obtain ⟨p, hp, hp'⟩ := h
-  refine' ⟨(p.map (_root_.algebraMap R A)).scaleRoots x, _, _⟩
-  · rw [Polynomial.monic_scaleRoots_iff]
-    exact hp.map _
-  convert Polynomial.scaleRoots_eval₂_mul (R := A ⊗[R] B) (S := A)
-      Algebra.TensorProduct.includeLeftRingHom (?_) x
-  any_goals exact 1 ⊗ₜ y
-  · simp only [Algebra.TensorProduct.includeLeftRingHom_apply,
-      Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul]
-  · simp only [Algebra.TensorProduct.includeLeftRingHom_apply, Algebra.TensorProduct.tmul_pow,
-      one_pow]
-    convert (mul_zero (M₀ := A ⊗[R] B) _).symm
-    erw [Polynomial.eval₂_map, Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap,
-      ← Polynomial.eval₂_map]
-    convert Polynomial.eval₂_at_apply
-      (Algebra.TensorProduct.includeRight : B →ₐ[R] A ⊗[R] B).toRingHom y
-    rw [Polynomial.eval_map, hp', _root_.map_zero]
+  rw [← mul_one x, ← smul_eq_mul, ← smul_tmul']
+  exact smul _ (h.map_of_comp_eq (algebraMap R A)
+    (Algebra.TensorProduct.includeRight (R := R) (A := A) (B := B)).toRingHom
+    Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap)
 #align is_integral.tmul IsIntegral.tmul
 
 section
@@ -801,7 +646,7 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
     · apply_fun fun q => coeff q p.natDegree at hp
       rw [coeff_map, coeff_zero, coeff_natDegree] at hp
       rw [hp, zero_mul]
-      exact f.is_integral_zero
+      exact f.isIntegralElem_zero
     · rw [Nat.one_le_iff_ne_zero, Classical.not_not] at h'
       rw [eq_C_of_natDegree_eq_zero h', eval₂_C] at h
       suffices p.map f = 0 by exact (hp this).elim
@@ -850,7 +695,7 @@ variable {R A B : Type*} [CommRing R] [CommRing A] [CommRing B]
 
 variable [Algebra R B] [Algebra A B] [IsIntegralClosure A R B]
 
-variable (R) (B) -- porting note: `{A}` was a `redundant binder annotation update`
+variable (R B)
 
 protected theorem isIntegral [Algebra R A] [IsScalarTower R A B] (x : A) : IsIntegral R x :=
   (isIntegral_algebraMap_iff (algebraMap_injective A R B)).mp <|
@@ -893,13 +738,13 @@ theorem mk'_zero (h : IsIntegral R (0 : B) := isIntegral_zero) : mk' A 0 h = 0 :
 
 -- Porting note: Left-hand side does not simplify @[simp]
 theorem mk'_add (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
-    mk' A (x + y) (IsIntegral.add hx hy) = mk' A x hx + mk' A y hy :=
+    mk' A (x + y) (hx.add hy) = mk' A x hx + mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebraMap_mk', RingHom.map_add]
 #align is_integral_closure.mk'_add IsIntegralClosure.mk'_add
 
 -- Porting note: Left-hand side does not simplify @[simp]
 theorem mk'_mul (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
-    mk' A (x * y) (IsIntegral.mul hx hy) = mk' A x hx * mk' A y hy :=
+    mk' A (x * y) (hx.mul hy) = mk' A x hx * mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebraMap_mk', RingHom.map_mul]
 #align is_integral_closure.mk'_mul IsIntegralClosure.mk'_mul
 
@@ -912,7 +757,6 @@ theorem mk'_algebraMap [Algebra R A] [IsScalarTower R A B] (x : R)
 
 section lift
 
--- porting note: `R` and `A` were redundant binder updates
 variable (B) {S : Type*} [CommRing S] [Algebra R S]
 -- split from above, since otherwise it does not synthesize `Semiring S`
 variable [Algebra S B] [IsScalarTower R S B]
@@ -947,14 +791,8 @@ variable [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A'
 /-- Integral closures are all isomorphic to each other. -/
 noncomputable def equiv : A ≃ₐ[R] A' :=
   AlgEquiv.ofAlgHom (lift _ B (isIntegral_algebra R B)) (lift _ B (isIntegral_algebra R B))
-    (by
-      ext x
-      apply algebraMap_injective A' R B
-      simp)
-    (by
-      ext x
-      apply algebraMap_injective A R B
-      simp)
+    (by ext x; apply algebraMap_injective A' R B; simp)
+    (by ext x; apply algebraMap_injective A R B; simp)
 #align is_integral_closure.equiv IsIntegralClosure.equiv
 
 @[simp]
@@ -974,36 +812,10 @@ open Algebra
 
 variable {R A B S T : Type*}
 
-variable [CommRing R] [CommRing A] [CommRing B] [CommRing S] [CommRing T]
+variable [CommRing R] [CommRing A] [Ring B] [CommRing S] [CommRing T]
 
 variable [Algebra A B] [Algebra R B] (f : R →+* S) (g : S →+* T)
 
-theorem isIntegral_trans_aux (x : B) {p : A[X]} (pmonic : Monic p) (hp : aeval x p = 0) :
-    IsIntegral (adjoin R (↑(p.map <| algebraMap A B).frange : Set B)) x := by
-  generalize hS : (↑(p.map <| algebraMap A B).frange : Set B) = S
-  have coeffs_mem : ∀ i, (p.map <| algebraMap A B).coeff i ∈ adjoin R S := by
-    intro i
-    by_cases hi : (p.map <| algebraMap A B).coeff i = 0
-    · rw [hi]
-      exact Subalgebra.zero_mem _
-    rw [← hS]
-    exact subset_adjoin (coeff_mem_frange _ _ hi)
-  obtain ⟨q, hq⟩ :
-    ∃ q : (adjoin R S)[X], q.map (algebraMap (adjoin R S) B) = (p.map <| algebraMap A B) := by
-    rw [← Set.mem_range]
-    exact (Polynomial.mem_map_range _).2 fun i => ⟨⟨_, coeffs_mem i⟩, rfl⟩
-  use q
-  constructor
-  · suffices h : (q.map (algebraMap (adjoin R S) B)).Monic
-    · refine' monic_of_injective _ h
-      exact Subtype.val_injective
-    · rw [hq]
-      exact pmonic.map _
-  · convert hp using 1
-    replace hq := congr_arg (eval x) hq
-    convert hq using 1 <;> symm <;> apply eval_map
-#align is_integral_trans_aux isIntegral_trans_aux
-
 variable [Algebra R A] [IsScalarTower R A B]
 
 /-- If A is an R-algebra all of whose elements are integral over R,
@@ -1011,97 +823,91 @@ and x is an element of an A-algebra that is integral over A, then x is integral
 theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegral A x) :
     IsIntegral R x := by
   rcases hx with ⟨p, pmonic, hp⟩
-  let S : Set B := ↑(p.map <| algebraMap A B).frange
-  refine' IsIntegral.of_mem_of_fg (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
-  refine' fg_trans (fg_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
-  · rw [Finset.mem_coe, frange, Finset.mem_image] at hx
-    rcases hx with ⟨i, _, rfl⟩
-    rw [coeff_map]
-    exact IsIntegral.map (IsScalarTower.toAlgHom R A B) (A_int _)
-  · apply IsIntegral.fg_adjoin_singleton
-    exact isIntegral_trans_aux _ pmonic hp
+  let S := adjoin R (p.frange : Set A)
+  have : Module.Finite R S := ⟨(Subalgebra.toSubmodule S).fg_top.mpr <|
+    fg_adjoin_of_finite p.frange.finite_toSet fun a _ ↦ A_int a⟩
+  let p' : S[X] := p.toSubring S.toSubring subset_adjoin
+  have hSx : IsIntegral S x := ⟨p', (p.monic_toSubring _ _).mpr pmonic, by
+    rw [IsScalarTower.algebraMap_eq S A B, ← eval₂_map]
+    convert hp; apply p.map_toSubring S.toSubring⟩
+  let Sx := Subalgebra.toSubmodule (adjoin S {x})
+  let MSx : Module S Sx := SMulMemClass.toModule _ -- the next line times out without this
+  have : Module.Finite S Sx := ⟨(Submodule.fg_top _).mpr hSx.fg_adjoin_singleton⟩
+  refine .of_mem_of_fg ((adjoin S {x}).restrictScalars R) ?_ _
+    ((Subalgebra.mem_restrictScalars R).mpr <| subset_adjoin rfl)
+  rw [← Submodule.fg_top, ← Module.finite_def]
+  letI : SMul S Sx := { MSx with } -- need this even though MSx is there
+  have : IsScalarTower R S Sx :=
+    Submodule.isScalarTower Sx -- Lean looks for `Module A Sx` without this
+  exact Module.Finite.trans S Sx
 #align is_integral_trans isIntegral_trans
+#noalign is_integral_trans_aux
 
 /-- If A is an R-algebra all of whose elements are integral over R,
 and B is an A-algebra all of whose elements are integral over A,
 then all elements of B are integral over R.-/
-nonrec theorem Algebra.isIntegral_trans
+protected theorem Algebra.IsIntegral.trans
     (hA : Algebra.IsIntegral R A) (hB : Algebra.IsIntegral A B) : Algebra.IsIntegral R B :=
-  fun x => isIntegral_trans hA x (hB x)
-#align algebra.is_integral_trans Algebra.isIntegral_trans
-
-theorem RingHom.isIntegral_trans (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.comp f).IsIntegral :=
-  @Algebra.isIntegral_trans R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
-    (@IsScalarTower.of_algebraMap_eq R S T _ _ _ f.toAlgebra g.toAlgebra (g.comp f).toAlgebra
-      (RingHom.comp_apply g f))
-    hf hg
-#align ring_hom.is_integral_trans RingHom.isIntegral_trans
-
-theorem RingHom.isIntegral_of_surjective (hf : Function.Surjective f) : f.IsIntegral := fun x =>
-  (hf x).recOn fun _y hy => (hy ▸ f.is_integral_map : f.IsIntegralElem x)
+  fun x ↦ isIntegral_trans hA x (hB x)
+#align algebra.is_integral_trans Algebra.IsIntegral.trans
+
+protected theorem RingHom.IsIntegral.trans
+    (hf : f.IsIntegral) (hg : g.IsIntegral) : (g.comp f).IsIntegral :=
+  letI := f.toAlgebra; letI := g.toAlgebra; letI := (g.comp f).toAlgebra
+  haveI : IsScalarTower R S T := IsScalarTower.of_algebraMap_eq fun _ ↦ rfl
+  Algebra.IsIntegral.trans hf hg
+#align ring_hom.is_integral_trans RingHom.IsIntegral.trans
+
+theorem RingHom.isIntegral_of_surjective (hf : Function.Surjective f) : f.IsIntegral :=
+  fun x ↦ (hf x).recOn fun _y hy ↦ hy ▸ f.isIntegralElem_map
 #align ring_hom.is_integral_of_surjective RingHom.isIntegral_of_surjective
 
-theorem isIntegral_of_surjective (h : Function.Surjective (algebraMap R A)) :
+theorem Algebra.isIntegral_of_surjective (h : Function.Surjective (algebraMap R A)) :
     Algebra.IsIntegral R A :=
   (algebraMap R A).isIntegral_of_surjective h
-#align is_integral_of_surjective isIntegral_of_surjective
+#align is_integral_of_surjective Algebra.isIntegral_of_surjective
 
 /-- If `R → A → B` is an algebra tower with `A → B` injective,
 then if the entire tower is an integral extension so is `R → A` -/
 theorem IsIntegral.tower_bot (H : Function.Injective (algebraMap A B)) {x : A}
-    (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x := by
-  rcases h with ⟨p, ⟨hp, hp'⟩⟩
-  refine' ⟨p, ⟨hp, _⟩⟩
-  rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map, eval₂_hom, ←
-    RingHom.map_zero (algebraMap A B)] at hp'
-  rw [eval₂_eq_eval_map]
-  exact H hp'
+    (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
+  (isIntegral_algHom_iff (IsScalarTower.toAlgHom R A B) H).mp h
 #align is_integral_tower_bot_of_is_integral IsIntegral.tower_bot
 
-nonrec theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injective g)
-    (hfg : (g.comp f).IsIntegral) : f.IsIntegral := fun x =>
-  @IsIntegral.tower_bot R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
-    (@IsScalarTower.of_algebraMap_eq R S T _ _ _ f.toAlgebra g.toAlgebra (g.comp f).toAlgebra
-      (RingHom.comp_apply g f))
-    hg x (hfg (g x))
-#align ring_hom.is_integral_tower_bot_of_is_integral RingHom.isIntegral_tower_bot_of_isIntegral
+nonrec theorem RingHom.IsIntegral.tower_bot (hg : Function.Injective g)
+    (hfg : (g.comp f).IsIntegral) : f.IsIntegral :=
+  letI := f.toAlgebra; letI := g.toAlgebra; letI := (g.comp f).toAlgebra
+  haveI : IsScalarTower R S T := IsScalarTower.of_algebraMap_eq fun _ ↦ rfl
+  fun x ↦ IsIntegral.tower_bot hg (hfg (g x))
+#align ring_hom.is_integral_tower_bot_of_is_integral RingHom.IsIntegral.tower_bot
 
 theorem IsIntegral.tower_bot_of_field {R A B : Type*} [CommRing R] [Field A]
     [CommRing B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]
     {x : A} (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
-  IsIntegral.tower_bot (algebraMap A B).injective h
+  h.tower_bot (algebraMap A B).injective
 #align is_integral_tower_bot_of_is_integral_field IsIntegral.tower_bot_of_field
 
-theorem RingHom.isIntegralElem_of_isIntegralElem_comp {x : T} (h : (g.comp f).IsIntegralElem x) :
+theorem RingHom.isIntegralElem.of_comp {x : T} (h : (g.comp f).IsIntegralElem x) :
     g.IsIntegralElem x :=
-  let ⟨p, ⟨hp, hp'⟩⟩ := h
+  let ⟨p, hp, hp'⟩ := h
   ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp'⟩
-#align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem_of_isIntegralElem_comp
+#align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem.of_comp
 
-theorem RingHom.isIntegral_tower_top_of_isIntegral (h : (g.comp f).IsIntegral) : g.IsIntegral :=
-  fun x => RingHom.isIntegralElem_of_isIntegralElem_comp f g (h x)
-#align ring_hom.is_integral_tower_top_of_is_integral RingHom.isIntegral_tower_top_of_isIntegral
+theorem RingHom.IsIntegral.tower_top (h : (g.comp f).IsIntegral) : g.IsIntegral :=
+  fun x ↦ RingHom.isIntegralElem.of_comp f g (h x)
+#align ring_hom.is_integral_tower_top_of_is_integral RingHom.IsIntegral.tower_top
 
-/-- If `R → A → B` is an algebra tower,
-then if the entire tower is an integral extension so is `A → B`. -/
-theorem isIntegral_tower_top_of_isIntegral {x : B} (h : IsIntegral R x) : IsIntegral A x := by
-  rcases h with ⟨p, ⟨hp, hp'⟩⟩
-  refine' ⟨p.map (algebraMap R A), ⟨hp.map (algebraMap R A), _⟩⟩
-  rw [IsScalarTower.algebraMap_eq R A B, ← eval₂_map] at hp'
-  exact hp'
-#align is_integral_tower_top_of_is_integral isIntegral_tower_top_of_isIntegral
-
-theorem RingHom.isIntegral_quotient_of_isIntegral {I : Ideal S} (hf : f.IsIntegral) :
+theorem RingHom.IsIntegral.quotient {I : Ideal S} (hf : f.IsIntegral) :
     (Ideal.quotientMap I f le_rfl).IsIntegral := by
   rintro ⟨x⟩
-  obtain ⟨p, ⟨p_monic, hpx⟩⟩ := hf x
-  refine' ⟨p.map (Ideal.Quotient.mk _), ⟨p_monic.map _, _⟩⟩
+  obtain ⟨p, p_monic, hpx⟩ := hf x
+  refine ⟨p.map (Ideal.Quotient.mk _), p_monic.map _, ?_⟩
   simpa only [hom_eval₂, eval₂_map] using congr_arg (Ideal.Quotient.mk I) hpx
-#align ring_hom.is_integral_quotient_of_is_integral RingHom.isIntegral_quotient_of_isIntegral
+#align ring_hom.is_integral_quotient_of_is_integral RingHom.IsIntegral.quotient
 
 theorem Algebra.IsIntegral.quotient {I : Ideal A} (hRA : Algebra.IsIntegral R A) :
     Algebra.IsIntegral (R ⧸ I.comap (algebraMap R A)) (A ⧸ I) :=
-  (algebraMap R A).isIntegral_quotient_of_isIntegral hRA
+  RingHom.IsIntegral.quotient (algebraMap R A) hRA
 #align is_integral_quotient_of_is_integral Algebra.IsIntegral.quotient
 
 theorem isIntegral_quotientMap_iff {I : Ideal S} :
@@ -1111,68 +917,43 @@ theorem isIntegral_quotientMap_iff {I : Ideal S} :
   -- Porting note: added type ascription
   have : (Ideal.quotientMap I f le_rfl).comp g = (Ideal.Quotient.mk I).comp f :=
     Ideal.quotientMap_comp_mk le_rfl
-  refine' ⟨fun h => _, fun h => RingHom.isIntegral_tower_top_of_isIntegral g _ (this ▸ h)⟩
-  refine' this ▸ RingHom.isIntegral_trans g (Ideal.quotientMap I f le_rfl) _ h
-  exact RingHom.isIntegral_of_surjective g Ideal.Quotient.mk_surjective
+  refine' ⟨fun h => _, fun h => RingHom.IsIntegral.tower_top g _ (this ▸ h)⟩
+  refine' this ▸ RingHom.IsIntegral.trans g (Ideal.quotientMap I f le_rfl) _ h
+  exact g.isIntegral_of_surjective Ideal.Quotient.mk_surjective
 #align is_integral_quotient_map_iff isIntegral_quotientMap_iff
 
 /-- If the integral extension `R → S` is injective, and `S` is a field, then `R` is also a field. -/
-theorem isField_of_isIntegral_of_isField {R S : Type*} [CommRing R] [Nontrivial R] [CommRing S]
-    [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S)
+theorem isField_of_isIntegral_of_isField {R S : Type*} [CommRing R] [CommRing S]
+    [Algebra R S] (H : Algebra.IsIntegral R S)
     (hRS : Function.Injective (algebraMap R S)) (hS : IsField S) : IsField R := by
-  refine' ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun {a} ha => _⟩
+  have := hS.nontrivial; have := Module.nontrivial R S
+  refine ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun {a} ha ↦ ?_⟩
   -- Let `a_inv` be the inverse of `algebraMap R S a`,
   -- then we need to show that `a_inv` is of the form `algebraMap R S b`.
-  obtain ⟨a_inv, ha_inv⟩ := hS.mul_inv_cancel
-    fun h => ha (hRS (_root_.trans h (RingHom.map_zero _).symm))
+  obtain ⟨a_inv, ha_inv⟩ := hS.mul_inv_cancel fun h ↦ ha (hRS (h.trans (RingHom.map_zero _).symm))
+  letI : Invertible a_inv := (Units.mkOfMulEqOne a_inv _ <| mul_comm _ a_inv ▸ ha_inv).invertible
   -- Let `p : R[X]` be monic with root `a_inv`,
-  -- and `q` be `p` with coefficients reversed (so `q(a) = q'(a) * a + 1`).
-  -- We claim that `q(a) = 0`, so `-q'(a)` is the inverse of `a`.
   obtain ⟨p, p_monic, hp⟩ := H a_inv
-  use -∑ i : ℕ in Finset.range p.natDegree, p.coeff i * a ^ (p.natDegree - i - 1)
-  -- `q(a) = 0`, because multiplying everything with `a_inv^n` gives `p(a_inv) = 0`.
-  -- TODO: this could be a lemma for `Polynomial.reverse`.
-  have hq : (∑ i : ℕ in Finset.range (p.natDegree + 1), p.coeff i * a ^ (p.natDegree - i)) = 0 := by
-    apply (injective_iff_map_eq_zero (algebraMap R S)).mp hRS
-    have a_inv_ne_zero : a_inv ≠ 0 := right_ne_zero_of_mul (mt ha_inv.symm.trans one_ne_zero)
-    refine' (mul_eq_zero.mp _).resolve_right (pow_ne_zero p.natDegree a_inv_ne_zero)
-    rw [eval₂_eq_sum_range] at hp
-    rw [map_sum, Finset.sum_mul]
-    refine' (Finset.sum_congr rfl fun i hi => _).trans hp
-    rw [RingHom.map_mul, mul_assoc]
-    congr
-    have : a_inv ^ p.natDegree = a_inv ^ (p.natDegree - i) * a_inv ^ i := by
-      rw [← pow_add a_inv, tsub_add_cancel_of_le (Nat.le_of_lt_succ (Finset.mem_range.mp hi))]
-    rw [RingHom.map_pow, this, ← mul_assoc, ← mul_pow, ha_inv, one_pow, one_mul]
-  -- Since `q(a) = 0` and `q(a) = q'(a) * a + 1`, we have `a * -q'(a) = 1`.
-  -- TODO: we could use a lemma for `Polynomial.divX` here.
-  rw [Finset.sum_range_succ_comm, p_monic.coeff_natDegree, one_mul, tsub_self, pow_zero,
-    add_eq_zero_iff_eq_neg, eq_comm] at hq
-  rw [mul_comm, neg_mul, Finset.sum_mul]
-  convert hq using 2
-  refine' Finset.sum_congr rfl fun i hi => _
-  have : 1 ≤ p.natDegree - i := le_tsub_of_add_le_left (Finset.mem_range.mp hi)
-  rw [mul_assoc, ← pow_succ', tsub_add_cancel_of_le this]
+  -- and `q` be `p` with coefficients reversed (so `q(a) = q'(a) * a + 1`).
+  -- We have `q(a) = 0`, so `-q'(a)` is the inverse of `a`.
+  use -p.reverse.divX.eval a -- -q'(a)
+  nth_rewrite 1 [mul_neg, ← eval_X (x := a), ← eval_mul, ← p_monic, ← coeff_zero_reverse,
+    ← add_eq_zero_iff_neg_eq, ← eval_C (a := p.reverse.coeff 0), ← eval_add, X_mul_divX_add,
+    ← (injective_iff_map_eq_zero' _).mp hRS, ← aeval_algebraMap_apply_eq_algebraMap_eval]
+  rwa [← eval₂_reverse_eq_zero_iff] at hp
 #align is_field_of_is_integral_of_is_field isField_of_isIntegral_of_isField
 
 theorem isField_of_isIntegral_of_isField' {R S : Type*} [CommRing R] [CommRing S] [IsDomain S]
     [Algebra R S] (H : Algebra.IsIntegral R S) (hR : IsField R) : IsField S := by
+  refine ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun {x} hx ↦ ?_⟩
+  have : Module.Finite R (adjoin R {x}) := ⟨(Submodule.fg_top _).mpr (H x).fg_adjoin_singleton⟩
   letI := hR.toField
-  refine' ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun {x} hx => _⟩
-  let A := Algebra.adjoin R ({x} : Set S)
-  haveI : IsNoetherian R A :=
-    isNoetherian_of_fg_of_noetherian (Subalgebra.toSubmodule A)
-      (IsIntegral.fg_adjoin_singleton x (H x))
-  haveI : Module.Finite R A := Module.IsNoetherian.finite R A
-  obtain ⟨y, hy⟩ :=
-    LinearMap.surjective_of_injective
-      (@LinearMap.mulLeft_injective R A _ _ _ _ ⟨x, subset_adjoin (Set.mem_singleton x)⟩ fun h =>
-        hx (Subtype.ext_iff.mp h))
-      1
+  obtain ⟨y, hy⟩ := FiniteDimensional.exists_mul_eq_one R
+    (K := adjoin R {x}) (x := ⟨x, subset_adjoin rfl⟩) (mt Subtype.ext_iff.mp hx)
   exact ⟨y, Subtype.ext_iff.mp hy⟩
 #align is_field_of_is_integral_of_is_field' isField_of_isIntegral_of_isField'
 
-theorem Algebra.IsIntegral.isField_iff_isField {R S : Type*} [CommRing R] [Nontrivial R]
+theorem Algebra.IsIntegral.isField_iff_isField {R S : Type*} [CommRing R]
     [CommRing S] [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S)
     (hRS : Function.Injective (algebraMap R S)) : IsField R ↔ IsField S :=
   ⟨isField_of_isIntegral_of_isField' H, isField_of_isIntegral_of_isField H hRS⟩
@@ -1180,14 +961,11 @@ theorem Algebra.IsIntegral.isField_iff_isField {R S : Type*} [CommRing R] [Nontr
 
 end Algebra
 
-theorem integralClosure_idem {R : Type*} {A : Type*} [CommRing R] [CommRing A] [Algebra R A] :
+theorem integralClosure_idem {R A : Type*} [CommRing R] [CommRing A] [Algebra R A] :
     integralClosure (integralClosure R A : Set A) A = ⊥ :=
-  eq_bot_iff.2 fun x hx =>
-    Algebra.mem_bot.2
-      ⟨⟨x,
-          @isIntegral_trans _ _ _ _ _ _ _ _ (integralClosure R A).algebra _
-            integralClosure.isIntegral x hx⟩,
-        rfl⟩
+  letI := (integralClosure R A).algebra
+  eq_bot_iff.2 fun x hx ↦ Algebra.mem_bot.2
+    ⟨⟨x, isIntegral_trans (A := integralClosure R A) integralClosure.isIntegral x hx⟩, rfl⟩
 #align integral_closure_idem integralClosure_idem
 
 section IsDomain

This PR tests a string-based tool for renaming declarations.

Inspired by this Zulip thread, I am trying to reduce the diff of #8406.

This PR makes the following renames:

| From | To |

@@ -121,7 +121,7 @@ variable [Field K] [Ring A] [Algebra K A] [FiniteDimensional K A]
 
 variable (K)
 
-theorem isIntegral_of_finite (e : A) : IsIntegral K e :=
+theorem IsIntegral.of_finite (e : A) : IsIntegral K e :=
   isIntegral_of_noetherian (IsNoetherian.iff_fg.2 inferInstance) _
 
 variable (A)
@@ -141,16 +141,16 @@ variable [CommRing R] [CommRing A] [CommRing B] [CommRing S]
 
 variable [Algebra R A] [Algebra R B] (f : R →+* S)
 
-theorem map_isIntegral {B C F : Type*} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
+theorem IsIntegral.map {B C F : Type*} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
     [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B} [AlgHomClass F A B C] (f : F)
     (hb : IsIntegral R b) : IsIntegral R (f b) := by
   obtain ⟨P, hP⟩ := hb
   refine' ⟨P, hP.1, _⟩
   rw [← aeval_def, show (aeval (f b)) P = (aeval (f b)) (P.map (algebraMap R A)) by simp,
     aeval_algHom_apply, aeval_map_algebraMap, aeval_def, hP.2, _root_.map_zero]
-#align map_is_integral map_isIntegral
+#align map_is_integral IsIntegral.map
 
-theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type*} [CommRing R] [CommRing S]
+theorem IsIntegral.map_of_comp_eq {R S T U : Type*} [CommRing R] [CommRing S]
     [CommRing T] [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
     (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
     IsIntegral T (ψ a) := by
@@ -159,11 +159,11 @@ theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type*} [CommRing R] [
   refine' ⟨p.map φ, hp.left.map _, _⟩
   rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply, eval_map, hp.right,
     RingHom.map_zero]
-#align is_integral_map_of_comp_eq_of_is_integral isIntegral_map_of_comp_eq_of_isIntegral
+#align is_integral_map_of_comp_eq_of_is_integral IsIntegral.map_of_comp_eq
 
 theorem isIntegral_algHom_iff {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A →ₐ[R] B) (hf : Function.Injective f) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x := by
-  refine' ⟨_, map_isIntegral f⟩
+  refine' ⟨_, IsIntegral.map f⟩
   rintro ⟨p, hp, hx⟩
   use p, hp
   rwa [← f.comp_algebraMap, ← AlgHom.coe_toRingHom, ← Polynomial.hom_eval₂, AlgHom.coe_toRingHom,
@@ -173,7 +173,7 @@ theorem isIntegral_algHom_iff {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Alg
 @[simp]
 theorem isIntegral_algEquiv {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
-  ⟨fun h => by simpa using map_isIntegral f.symm.toAlgHom h, map_isIntegral f.toAlgHom⟩
+  ⟨fun h => by simpa using IsIntegral.map f.symm.toAlgHom h, IsIntegral.map f.toAlgHom⟩
 #align is_integral_alg_equiv isIntegral_algEquiv
 
 theorem isIntegral_of_isScalarTower [Algebra A B] [IsScalarTower R A B] {x : B}
@@ -184,14 +184,15 @@ theorem isIntegral_of_isScalarTower [Algebra A B] [IsScalarTower R A B] {x : B}
 
 theorem map_isIntegral_int {B C F : Type*} [Ring B] [Ring C] {b : B} [RingHomClass F B C] (f : F)
     (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
-  map_isIntegral (f : B →+* C).toIntAlgHom hb
+  IsIntegral.map (f : B →+* C).toIntAlgHom hb
 #align map_is_integral_int map_isIntegral_int
 
-theorem isIntegral_ofSubring {x : A} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
+theorem IsIntegral.of_subring {x : A} (T : Subring R) (hx : IsIntegral T x) : IsIntegral R x :=
   isIntegral_of_isScalarTower hx
-#align is_integral_of_subring isIntegral_ofSubring
+#align is_integral_of_subring IsIntegral.of_subring
 
-theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A} (h : IsIntegral R x) :
+protected theorem IsIntegral.algebraMap [Algebra A B] [IsScalarTower R A B] {x : A}
+    (h : IsIntegral R x) :
     IsIntegral R (algebraMap A B x) := by
   rcases h with ⟨f, hf, hx⟩
   use f, hf
@@ -211,10 +212,10 @@ theorem isIntegral_iff_isIntegral_closure_finite {r : A} :
     refine' ⟨_, Finset.finite_toSet _, p.restriction, monic_restriction.2 hmp, _⟩
     rw [← aeval_def, ← aeval_map_algebraMap R r p.restriction, map_restriction, aeval_def, hpr]
   rcases hr with ⟨s, _, hsr⟩
-  exact isIntegral_ofSubring _ hsr
+  exact IsIntegral.of_subring _ hsr
 #align is_integral_iff_is_integral_closure_finite isIntegral_iff_isIntegral_closure_finite
 
-theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
+theorem IsIntegral.fg_adjoin_singleton (x : A) (hx : IsIntegral R x) :
     (Algebra.adjoin R ({x} : Set A)).toSubmodule.FG := by
   rcases hx with ⟨f, hfm, hfx⟩
   exists Finset.image ((· ^ ·) x) (Finset.range (natDegree f + 1))
@@ -242,9 +243,9 @@ theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rw [degree_le_iff_coeff_zero] at this
   rw [mem_support_iff] at hkq; apply hkq; apply this
   exact lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 hk)
-#align fg_adjoin_singleton_of_integral FG_adjoin_singleton_of_integral
+#align fg_adjoin_singleton_of_integral IsIntegral.fg_adjoin_singleton
 
-theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
+theorem fg_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsIntegral R x) :
     (Algebra.adjoin R s).toSubmodule.FG :=
   Set.Finite.induction_on hfs
     (fun _ =>
@@ -255,18 +256,18 @@ theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
       rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule]
       exact
         FG.mul (ih fun i hi => his i <| Set.mem_insert_of_mem a hi)
-          (FG_adjoin_singleton_of_integral _ <| his a <| Set.mem_insert a s))
+          (IsIntegral.fg_adjoin_singleton _ <| his a <| Set.mem_insert a s))
     his
-#align fg_adjoin_of_finite FG_adjoin_of_finite
+#align fg_adjoin_of_finite fg_adjoin_of_finite
 
 theorem isNoetherian_adjoin_finset [IsNoetherianRing R] (s : Finset A)
     (hs : ∀ x ∈ s, IsIntegral R x) : IsNoetherian R (Algebra.adjoin R (↑s : Set A)) :=
-  isNoetherian_of_fg_of_noetherian _ (FG_adjoin_of_finite s.finite_toSet hs)
+  isNoetherian_of_fg_of_noetherian _ (fg_adjoin_of_finite s.finite_toSet hs)
 #align is_noetherian_adjoin_finset isNoetherian_adjoin_finset
 
 /-- If `S` is a sub-`R`-algebra of `A` and `S` is finitely-generated as an `R`-module,
   then all elements of `S` are integral over `R`. -/
-theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) :
+theorem IsIntegral.of_mem_of_fg (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x : A) (hx : x ∈ S) :
     IsIntegral R x := by
   -- say `x ∈ S`. We want to prove that `x` is integral over `R`.
   -- Say `S` is generated as an `R`-module by the set `y`.
@@ -292,7 +293,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   let S₀ : Subring R :=
     Subring.closure ↑(lx.frange ∪ Finset.biUnion Finset.univ (Finsupp.frange ∘ ly))
   -- It suffices to prove that `x` is integral over `S₀`.
-  refine' isIntegral_ofSubring S₀ _
+  refine' IsIntegral.of_subring S₀ _
   letI : CommRing S₀ := SubringClass.toCommRing S₀
   letI : Algebra S₀ A := Algebra.ofSubring S₀
   -- Claim: the `S₀`-module span (in `A`) of the set `y ∪ {1}` is closed under
@@ -365,7 +366,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   change (⟨_, this⟩ : S₀) • r ∈ _
   rw [Finsupp.mem_supported] at hlx1
   exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _
-#align is_integral_of_mem_of_fg isIntegral_of_mem_of_FG
+#align is_integral_of_mem_of_fg IsIntegral.of_mem_of_fg
 
 theorem Module.End.isIntegral {M : Type*} [AddCommGroup M] [Module R M] [Module.Finite R M] :
     Algebra.IsIntegral R (Module.End R M) :=
@@ -419,7 +420,7 @@ variable {f}
 
 theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
   letI := f.toAlgebra
-  fun _ => isIntegral_of_mem_of_FG ⊤ h.1 _ trivial
+  fun _ => IsIntegral.of_mem_of_fg ⊤ h.1 _ trivial
 #align ring_hom.finite.to_is_integral RingHom.Finite.to_isIntegral
 
 alias RingHom.IsIntegral.of_finite := RingHom.Finite.to_isIntegral
@@ -431,7 +432,7 @@ theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.
   constructor
   change (⊤ : Subalgebra R S).toSubmodule.FG
   rw [← hs]
-  exact FG_adjoin_of_finite (Set.toFinite _) fun x _ => h x
+  exact fg_adjoin_of_finite (Set.toFinite _) fun x _ => h x
 #align ring_hom.is_integral.to_finite RingHom.IsIntegral.to_finite
 
 alias RingHom.Finite.of_isIntegral_of_finiteType := RingHom.IsIntegral.to_finite
@@ -459,7 +460,7 @@ theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.Fin
 #align algebra.is_integral.finite Algebra.IsIntegral.finite
 
 theorem Algebra.IsIntegral.of_finite [h : Module.Finite R A] : Algebra.IsIntegral R A :=
-  fun _ ↦ isIntegral_of_mem_of_FG ⊤ h.1 _ trivial
+  fun _ ↦ IsIntegral.of_mem_of_fg ⊤ h.1 _ trivial
 #align algebra.is_integral.of_finite Algebra.IsIntegral.of_finite
 
 /-- finite = integral + finite type -/
@@ -473,17 +474,17 @@ variable (f)
 theorem RingHom.is_integral_of_mem_closure {x y z : S} (hx : f.IsIntegralElem x)
     (hy : f.IsIntegralElem y) (hz : z ∈ Subring.closure ({x, y} : Set S)) : f.IsIntegralElem z := by
   letI : Algebra R S := f.toAlgebra
-  have := (FG_adjoin_singleton_of_integral x hx).mul (FG_adjoin_singleton_of_integral y hy)
+  have := (IsIntegral.fg_adjoin_singleton x hx).mul (IsIntegral.fg_adjoin_singleton y hy)
   rw [← Algebra.adjoin_union_coe_submodule, Set.singleton_union] at this
   exact
-    isIntegral_of_mem_of_FG (Algebra.adjoin R {x, y}) this z
+    IsIntegral.of_mem_of_fg (Algebra.adjoin R {x, y}) this z
       (Algebra.mem_adjoin_iff.2 <| Subring.closure_mono (Set.subset_union_right _ _) hz)
 #align ring_hom.is_integral_of_mem_closure RingHom.is_integral_of_mem_closure
 
-theorem isIntegral_of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
+theorem IsIntegral.of_mem_closure {x y z : A} (hx : IsIntegral R x) (hy : IsIntegral R y)
     (hz : z ∈ Subring.closure ({x, y} : Set A)) : IsIntegral R z :=
   (algebraMap R A).is_integral_of_mem_closure hx hy hz
-#align is_integral_of_mem_closure isIntegral_of_mem_closure
+#align is_integral_of_mem_closure IsIntegral.of_mem_closure
 
 theorem RingHom.is_integral_zero : f.IsIntegralElem 0 :=
   f.map_zero ▸ f.is_integral_map
@@ -507,28 +508,28 @@ theorem RingHom.is_integral_add {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIn
     Subring.add_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl))
 #align ring_hom.is_integral_add RingHom.is_integral_add
 
-theorem isIntegral_add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
+theorem IsIntegral.add {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x + y) :=
   (algebraMap R A).is_integral_add hx hy
-#align is_integral_add isIntegral_add
+#align is_integral_add IsIntegral.add
 
 theorem RingHom.is_integral_neg {x : S} (hx : f.IsIntegralElem x) : f.IsIntegralElem (-x) :=
   f.is_integral_of_mem_closure hx hx (Subring.neg_mem _ (Subring.subset_closure (Or.inl rfl)))
 #align ring_hom.is_integral_neg RingHom.is_integral_neg
 
-theorem isIntegral_neg {x : A} (hx : IsIntegral R x) : IsIntegral R (-x) :=
+theorem IsIntegral.neg {x : A} (hx : IsIntegral R x) : IsIntegral R (-x) :=
   (algebraMap R A).is_integral_neg hx
-#align is_integral_neg isIntegral_neg
+#align is_integral_neg IsIntegral.neg
 
 theorem RingHom.is_integral_sub {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x - y) := by
   simpa only [sub_eq_add_neg] using f.is_integral_add hx (f.is_integral_neg hy)
 #align ring_hom.is_integral_sub RingHom.is_integral_sub
 
-theorem isIntegral_sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
+theorem IsIntegral.sub {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x - y) :=
   (algebraMap R A).is_integral_sub hx hy
-#align is_integral_sub isIntegral_sub
+#align is_integral_sub IsIntegral.sub
 
 theorem RingHom.is_integral_mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIntegralElem y) :
     f.IsIntegralElem (x * y) :=
@@ -536,24 +537,24 @@ theorem RingHom.is_integral_mul {x y : S} (hx : f.IsIntegralElem x) (hy : f.IsIn
     (Subring.mul_mem _ (Subring.subset_closure (Or.inl rfl)) (Subring.subset_closure (Or.inr rfl)))
 #align ring_hom.is_integral_mul RingHom.is_integral_mul
 
-theorem isIntegral_mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
+theorem IsIntegral.mul {x y : A} (hx : IsIntegral R x) (hy : IsIntegral R y) :
     IsIntegral R (x * y) :=
   (algebraMap R A).is_integral_mul hx hy
-#align is_integral_mul isIntegral_mul
+#align is_integral_mul IsIntegral.mul
 
-theorem isIntegral_smul [Algebra S A] [Algebra R S] [IsScalarTower R S A] {x : A} (r : R)
+theorem IsIntegral.smul [Algebra S A] [Algebra R S] [IsScalarTower R S A] {x : A} (r : R)
     (hx : IsIntegral S x) : IsIntegral S (r • x) := by
   rw [Algebra.smul_def, IsScalarTower.algebraMap_apply R S A]
-  exact isIntegral_mul isIntegral_algebraMap hx
-#align is_integral_smul isIntegral_smul
+  exact IsIntegral.mul isIntegral_algebraMap hx
+#align is_integral_smul IsIntegral.smul
 
-theorem isIntegral_of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
+theorem IsIntegral.of_pow {x : A} {n : ℕ} (hn : 0 < n) (hx : IsIntegral R <| x ^ n) :
     IsIntegral R x := by
   rcases hx with ⟨p, ⟨hmonic, heval⟩⟩
   exact
     ⟨expand R n p, Monic.expand hn hmonic, by
       rwa [eval₂_eq_eval_map, map_expand, expand_eval, ← eval₂_eq_eval_map]⟩
-#align is_integral_of_pow isIntegral_of_pow
+#align is_integral_of_pow IsIntegral.of_pow
 
 variable (R A)
 
@@ -562,17 +563,17 @@ def integralClosure : Subalgebra R A where
   carrier := { r | IsIntegral R r }
   zero_mem' := isIntegral_zero
   one_mem' := isIntegral_one
-  add_mem' := isIntegral_add
-  mul_mem' := isIntegral_mul
+  add_mem' := IsIntegral.add
+  mul_mem' := IsIntegral.mul
   algebraMap_mem' _ := isIntegral_algebraMap
 #align integral_closure integralClosure
 
-theorem mem_integralClosure_iff_mem_FG {r : A} :
+theorem mem_integralClosure_iff_mem_fg {r : A} :
     r ∈ integralClosure R A ↔ ∃ M : Subalgebra R A, M.toSubmodule.FG ∧ r ∈ M :=
   ⟨fun hr =>
-    ⟨Algebra.adjoin R {r}, FG_adjoin_singleton_of_integral _ hr, Algebra.subset_adjoin rfl⟩,
-    fun ⟨M, Hf, hrM⟩ => isIntegral_of_mem_of_FG M Hf _ hrM⟩
-#align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_FG
+    ⟨Algebra.adjoin R {r}, IsIntegral.fg_adjoin_singleton _ hr, Algebra.subset_adjoin rfl⟩,
+    fun ⟨M, Hf, hrM⟩ => IsIntegral.of_mem_of_fg M Hf _ hrM⟩
+#align mem_integral_closure_iff_mem_fg mem_integralClosure_iff_mem_fg
 
 variable {R} {A}
 
@@ -604,9 +605,9 @@ theorem integralClosure_map_algEquiv (f : A ≃ₐ[R] B) :
   rw [Subalgebra.mem_map]
   constructor
   · rintro ⟨x, hx, rfl⟩
-    exact map_isIntegral f hx
+    exact IsIntegral.map f hx
   · intro hy
-    use f.symm y, map_isIntegral (f.symm : B →ₐ[R] A) hy
+    use f.symm y, IsIntegral.map (f.symm : B →ₐ[R] A) hy
     simp
 #align integral_closure_map_alg_equiv integralClosure_map_algEquiv
 
@@ -625,16 +626,16 @@ theorem RingHom.isIntegral_of_isIntegral_mul_unit (x y : S) (r : R) (hr : f r *
   rw [mul_comm x y, ← mul_assoc, hr, one_mul]
 #align ring_hom.is_integral_of_is_integral_mul_unit RingHom.isIntegral_of_isIntegral_mul_unit
 
-theorem isIntegral_of_isIntegral_mul_unit {x y : A} {r : R} (hr : algebraMap R A r * y = 1)
+theorem IsIntegral.of_mul_unit {x y : A} {r : R} (hr : algebraMap R A r * y = 1)
     (hx : IsIntegral R (x * y)) : IsIntegral R x :=
   (algebraMap R A).isIntegral_of_isIntegral_mul_unit x y r hr hx
-#align is_integral_of_is_integral_mul_unit isIntegral_of_isIntegral_mul_unit
+#align is_integral_of_is_integral_mul_unit IsIntegral.of_mul_unit
 
 /-- Generalization of `isIntegral_of_mem_closure` bootstrapped up from that lemma -/
 theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x) :
     ∀ x ∈ Subring.closure G, IsIntegral R x := fun _ hx =>
-  Subring.closure_induction hx hG isIntegral_zero isIntegral_one (fun _ _ => isIntegral_add)
-    (fun _ => isIntegral_neg) fun _ _ => isIntegral_mul
+  Subring.closure_induction hx hG isIntegral_zero isIntegral_one (fun _ _ => IsIntegral.add)
+    (fun _ => IsIntegral.neg) fun _ _ => IsIntegral.mul
 #align is_integral_of_mem_closure' isIntegral_of_mem_closure'
 
 theorem isIntegral_of_mem_closure'' {S : Type*} [CommRing S] {f : R →+* S} (G : Set S)
@@ -682,7 +683,7 @@ theorem IsIntegral.det {n : Type*} [Fintype n] [DecidableEq n] {M : Matrix n n A
 
 @[simp]
 theorem IsIntegral.pow_iff {x : A} {n : ℕ} (hn : 0 < n) : IsIntegral R (x ^ n) ↔ IsIntegral R x :=
-  ⟨isIntegral_of_pow hn, fun hx => IsIntegral.pow hx n⟩
+  ⟨IsIntegral.of_pow hn, fun hx => IsIntegral.pow hx n⟩
 #align is_integral.pow_iff IsIntegral.pow_iff
 
 open TensorProduct
@@ -892,13 +893,13 @@ theorem mk'_zero (h : IsIntegral R (0 : B) := isIntegral_zero) : mk' A 0 h = 0 :
 
 -- Porting note: Left-hand side does not simplify @[simp]
 theorem mk'_add (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
-    mk' A (x + y) (isIntegral_add hx hy) = mk' A x hx + mk' A y hy :=
+    mk' A (x + y) (IsIntegral.add hx hy) = mk' A x hx + mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebraMap_mk', RingHom.map_add]
 #align is_integral_closure.mk'_add IsIntegralClosure.mk'_add
 
 -- Porting note: Left-hand side does not simplify @[simp]
 theorem mk'_mul (x y : B) (hx : IsIntegral R x) (hy : IsIntegral R y) :
-    mk' A (x * y) (isIntegral_mul hx hy) = mk' A x hx * mk' A y hy :=
+    mk' A (x * y) (IsIntegral.mul hx hy) = mk' A x hx * mk' A y hy :=
   algebraMap_injective A R B <| by simp only [algebraMap_mk', RingHom.map_mul]
 #align is_integral_closure.mk'_mul IsIntegralClosure.mk'_mul
 
@@ -1011,13 +1012,13 @@ theorem isIntegral_trans (A_int : Algebra.IsIntegral R A) (x : B) (hx : IsIntegr
     IsIntegral R x := by
   rcases hx with ⟨p, pmonic, hp⟩
   let S : Set B := ↑(p.map <| algebraMap A B).frange
-  refine' isIntegral_of_mem_of_FG (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
-  refine' fg_trans (FG_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
+  refine' IsIntegral.of_mem_of_fg (adjoin R (S ∪ {x})) _ _ (subset_adjoin <| Or.inr rfl)
+  refine' fg_trans (fg_adjoin_of_finite (Finset.finite_toSet _) fun x hx => _) _
   · rw [Finset.mem_coe, frange, Finset.mem_image] at hx
     rcases hx with ⟨i, _, rfl⟩
     rw [coeff_map]
-    exact map_isIntegral (IsScalarTower.toAlgHom R A B) (A_int _)
-  · apply FG_adjoin_singleton_of_integral
+    exact IsIntegral.map (IsScalarTower.toAlgHom R A B) (A_int _)
+  · apply IsIntegral.fg_adjoin_singleton
     exact isIntegral_trans_aux _ pmonic hp
 #align is_integral_trans isIntegral_trans
 
@@ -1047,7 +1048,7 @@ theorem isIntegral_of_surjective (h : Function.Surjective (algebraMap R A)) :
 
 /-- If `R → A → B` is an algebra tower with `A → B` injective,
 then if the entire tower is an integral extension so is `R → A` -/
-theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A B)) {x : A}
+theorem IsIntegral.tower_bot (H : Function.Injective (algebraMap A B)) {x : A}
     (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x := by
   rcases h with ⟨p, ⟨hp, hp'⟩⟩
   refine' ⟨p, ⟨hp, _⟩⟩
@@ -1055,21 +1056,21 @@ theorem isIntegral_tower_bot_of_isIntegral (H : Function.Injective (algebraMap A
     RingHom.map_zero (algebraMap A B)] at hp'
   rw [eval₂_eq_eval_map]
   exact H hp'
-#align is_integral_tower_bot_of_is_integral isIntegral_tower_bot_of_isIntegral
+#align is_integral_tower_bot_of_is_integral IsIntegral.tower_bot
 
 nonrec theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injective g)
     (hfg : (g.comp f).IsIntegral) : f.IsIntegral := fun x =>
-  @isIntegral_tower_bot_of_isIntegral R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
+  @IsIntegral.tower_bot R S T _ _ _ g.toAlgebra (g.comp f).toAlgebra f.toAlgebra
     (@IsScalarTower.of_algebraMap_eq R S T _ _ _ f.toAlgebra g.toAlgebra (g.comp f).toAlgebra
       (RingHom.comp_apply g f))
     hg x (hfg (g x))
 #align ring_hom.is_integral_tower_bot_of_is_integral RingHom.isIntegral_tower_bot_of_isIntegral
 
-theorem isIntegral_tower_bot_of_isIntegral_field {R A B : Type*} [CommRing R] [Field A]
+theorem IsIntegral.tower_bot_of_field {R A B : Type*} [CommRing R] [Field A]
     [CommRing B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]
     {x : A} (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
-  isIntegral_tower_bot_of_isIntegral (algebraMap A B).injective h
-#align is_integral_tower_bot_of_is_integral_field isIntegral_tower_bot_of_isIntegral_field
+  IsIntegral.tower_bot (algebraMap A B).injective h
+#align is_integral_tower_bot_of_is_integral_field IsIntegral.tower_bot_of_field
 
 theorem RingHom.isIntegralElem_of_isIntegralElem_comp {x : T} (h : (g.comp f).IsIntegralElem x) :
     g.IsIntegralElem x :=
@@ -1098,10 +1099,10 @@ theorem RingHom.isIntegral_quotient_of_isIntegral {I : Ideal S} (hf : f.IsIntegr
   simpa only [hom_eval₂, eval₂_map] using congr_arg (Ideal.Quotient.mk I) hpx
 #align ring_hom.is_integral_quotient_of_is_integral RingHom.isIntegral_quotient_of_isIntegral
 
-theorem isIntegral_quotient_of_isIntegral {I : Ideal A} (hRA : Algebra.IsIntegral R A) :
+theorem Algebra.IsIntegral.quotient {I : Ideal A} (hRA : Algebra.IsIntegral R A) :
     Algebra.IsIntegral (R ⧸ I.comap (algebraMap R A)) (A ⧸ I) :=
   (algebraMap R A).isIntegral_quotient_of_isIntegral hRA
-#align is_integral_quotient_of_is_integral isIntegral_quotient_of_isIntegral
+#align is_integral_quotient_of_is_integral Algebra.IsIntegral.quotient
 
 theorem isIntegral_quotientMap_iff {I : Ideal S} :
     (Ideal.quotientMap I f le_rfl).IsIntegral ↔
@@ -1161,7 +1162,7 @@ theorem isField_of_isIntegral_of_isField' {R S : Type*} [CommRing R] [CommRing S
   let A := Algebra.adjoin R ({x} : Set S)
   haveI : IsNoetherian R A :=
     isNoetherian_of_fg_of_noetherian (Subalgebra.toSubmodule A)
-      (FG_adjoin_singleton_of_integral x (H x))
+      (IsIntegral.fg_adjoin_singleton x (H x))
   haveI : Module.Finite R A := Module.IsNoetherian.finite R A
   obtain ⟨y, hy⟩ :=
     LinearMap.surjective_of_injective

We'll need to do this step anyway when it is time to remove them all.

(See #8214 where I'm benchmarking the removal.)

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

@@ -1075,8 +1075,7 @@ theorem RingHom.isIntegralElem_of_isIntegralElem_comp {x : T} (h : (g.comp f).Is
     g.IsIntegralElem x :=
   let ⟨p, ⟨hp, hp'⟩⟩ := h
   ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp'⟩
-#align ring_hom.is_integral_elem_of_is_integral_elem_comp
-  RingHom.isIntegralElem_of_isIntegralElem_comp
+#align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem_of_isIntegralElem_comp
 
 theorem RingHom.isIntegral_tower_top_of_isIntegral (h : (g.comp f).IsIntegral) : g.IsIntegral :=
   fun x => RingHom.isIntegralElem_of_isIntegralElem_comp f g (h x)

@@ -728,7 +728,7 @@ theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ nat
   · simp [h₁]
   · rw [h₂, leadingCoeff, ← pow_succ, tsub_add_cancel_of_le hp]
   · rw [mul_assoc, ← pow_add, tsub_add_cancel_of_le]
-    apply Nat.le_pred_of_lt
+    apply Nat.le_sub_one_of_lt
     rw [lt_iff_le_and_ne]
     exact ⟨le_natDegree_of_ne_zero h₁, h₂⟩
 #align normalize_scale_roots_coeff_mul_leading_coeff_pow normalizeScaleRoots_coeff_mul_leadingCoeff_pow

Also golfs Normal.of_algEquiv and Algebra.IsIntegral.of_finite and refactors Algebra.IsAlgebraic.bijective_of_isScalarTower.

@@ -458,12 +458,8 @@ theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.Fin
   exact Algebra.smul_def _ _
 #align algebra.is_integral.finite Algebra.IsIntegral.finite
 
-theorem Algebra.IsIntegral.of_finite [h : Module.Finite R A] : Algebra.IsIntegral R A := by
-  apply RingHom.Finite.to_isIntegral
-  rw [RingHom.Finite]
-  convert h
-  ext
-  exact (Algebra.smul_def _ _).symm
+theorem Algebra.IsIntegral.of_finite [h : Module.Finite R A] : Algebra.IsIntegral R A :=
+  fun _ ↦ isIntegral_of_mem_of_FG ⊤ h.1 _ trivial
 #align algebra.is_integral.of_finite Algebra.IsIntegral.of_finite
 
 /-- finite = integral + finite type -/

And the same thing for StarSubalgebra R A. IntermediateField was already handled in #7957.

As a result, nine (obvious) lemmas are now true by definition.

This slightly adjusts the statement of Algebra.toSubmodule_bot to make it simpler and true by definition; the original statement can be recovered by rewriting by Submodule.one_eq_span, which I've had to add in some downstream proofs.

@@ -250,9 +250,7 @@ theorem FG_adjoin_of_finite {s : Set A} (hfs : s.Finite) (his : ∀ x ∈ s, IsI
     (fun _ =>
       ⟨{1},
         Submodule.ext fun x => by
-          erw [Algebra.adjoin_empty, Finset.coe_singleton, ← one_eq_span, one_eq_range,
-            LinearMap.mem_range, Algebra.mem_bot]
-          rfl⟩)
+          rw [Algebra.adjoin_empty, Finset.coe_singleton, ← one_eq_span, Algebra.toSubmodule_bot]⟩)
     (fun {a s} _ _ ih his => by
       rw [← Set.union_singleton, Algebra.adjoin_union_coe_submodule]
       exact

Removes nonterminal simps on lines looking like simp [...]

@@ -348,7 +348,7 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
       rw [Algebra.algebraMap_eq_smul_one]
       exact smul_mem (span S₀ (insert (1 : A) (y : Set A))) y' (subset_span (Or.inl rfl))
   have foo : ∀ z, z ∈ S₁ ↔ z ∈ Algebra.adjoin (↥S₀) (y : Set A)
-  simp [this]
+  simp only [this, Finset.univ_eq_attach, Subalgebra.mem_toSubring, forall_const]
   haveI : IsNoetherianRing S₀ := is_noetherian_subring_closure _ (Finset.finite_toSet _)
   refine'
     isIntegral_of_submodule_noetherian (Algebra.adjoin S₀ ↑y)

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

@@ -227,7 +227,7 @@ theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
     exact (Algebra.adjoin R {x}).pow_mem (Algebra.subset_adjoin (Set.mem_singleton _)) k
   intro r hr; change r ∈ Algebra.adjoin R ({x} : Set A) at hr
   rw [Algebra.adjoin_singleton_eq_range_aeval] at hr
-  rcases(aeval x).mem_range.mp hr with ⟨p, rfl⟩
+  rcases (aeval x).mem_range.mp hr with ⟨p, rfl⟩
   rw [← modByMonic_add_div p hfm]
   rw [← aeval_def] at hfx
   rw [AlgHom.map_add, AlgHom.map_mul, hfx, zero_mul, add_zero]

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -848,7 +848,7 @@ namespace IsIntegralClosure
 
 -- Porting note: added to work around missing infer kind support
 theorem algebraMap_injective (A R B : Type*) [CommRing R] [CommSemiring A] [CommRing B]
-  [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] : Function.Injective (algebraMap A B) :=
+    [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] : Function.Injective (algebraMap A B) :=
   algebraMap_injective' R
 
 variable {R A B : Type*} [CommRing R] [CommRing A] [CommRing B]

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>

@@ -1204,7 +1204,7 @@ instance : IsDomain (integralClosure R S) :=
   inferInstance
 
 theorem roots_mem_integralClosure {f : R[X]} (hf : f.Monic) {a : S}
-    (ha : a ∈ (f.map <| algebraMap R S).roots) : a ∈ integralClosure R S :=
+    (ha : a ∈ f.aroots S) : a ∈ integralClosure R S :=
   ⟨f, hf, (eval₂_eq_eval_map _).trans <| (mem_roots <| (hf.map _).ne_zero).1 ha⟩
 #align roots_mem_integral_closure roots_mem_integralClosure
 

@@ -424,7 +424,7 @@ theorem RingHom.Finite.to_isIntegral (h : f.Finite) : f.IsIntegral :=
   fun _ => isIntegral_of_mem_of_FG ⊤ h.1 _ trivial
 #align ring_hom.finite.to_is_integral RingHom.Finite.to_isIntegral
 
-alias RingHom.Finite.to_isIntegral ← RingHom.IsIntegral.of_finite
+alias RingHom.IsIntegral.of_finite := RingHom.Finite.to_isIntegral
 #align ring_hom.is_integral.of_finite RingHom.IsIntegral.of_finite
 
 theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.Finite := by
@@ -436,7 +436,7 @@ theorem RingHom.IsIntegral.to_finite (h : f.IsIntegral) (h' : f.FiniteType) : f.
   exact FG_adjoin_of_finite (Set.toFinite _) fun x _ => h x
 #align ring_hom.is_integral.to_finite RingHom.IsIntegral.to_finite
 
-alias RingHom.IsIntegral.to_finite ← RingHom.Finite.of_isIntegral_of_finiteType
+alias RingHom.Finite.of_isIntegral_of_finiteType := RingHom.IsIntegral.to_finite
 #align ring_hom.finite.of_is_integral_of_finite_type RingHom.Finite.of_isIntegral_of_finiteType
 
 /-- finite = integral + finite type -/

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

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

@@ -230,7 +230,7 @@ theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rcases(aeval x).mem_range.mp hr with ⟨p, rfl⟩
   rw [← modByMonic_add_div p hfm]
   rw [← aeval_def] at hfx
-  rw [AlgHom.map_add, AlgHom.map_mul, hfx, MulZeroClass.zero_mul, add_zero]
+  rw [AlgHom.map_add, AlgHom.map_mul, hfx, zero_mul, add_zero]
   have : degree (p %ₘ f) ≤ degree f := degree_modByMonic_le p hfm
   generalize p %ₘ f = q at this ⊢
   rw [← sum_C_mul_X_pow_eq q, aeval_def, eval₂_sum, sum_def]
@@ -707,7 +707,7 @@ theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x 
       Algebra.TensorProduct.tmul_mul_tmul, mul_one, one_mul]
   · simp only [Algebra.TensorProduct.includeLeftRingHom_apply, Algebra.TensorProduct.tmul_pow,
       one_pow]
-    convert (MulZeroClass.mul_zero (M₀ := A ⊗[R] B) _).symm
+    convert (mul_zero (M₀ := A ⊗[R] B) _).symm
     erw [Polynomial.eval₂_map, Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap,
       ← Polynomial.eval₂_map]
     convert Polynomial.eval₂_at_apply
@@ -729,7 +729,7 @@ theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ nat
     (normalizeScaleRoots p).coeff i * p.leadingCoeff ^ i =
       p.coeff i * p.leadingCoeff ^ (p.natDegree - 1) := by
   simp only [normalizeScaleRoots, finset_sum_coeff, coeff_monomial, Finset.sum_ite_eq', one_mul,
-    MulZeroClass.zero_mul, mem_support_iff, ite_mul, Ne.def, ite_not]
+    zero_mul, mem_support_iff, ite_mul, Ne.def, ite_not]
   split_ifs with h₁ h₂
   · simp [h₁]
   · rw [h₂, leadingCoeff, ← pow_succ, tsub_add_cancel_of_le hp]
@@ -801,11 +801,11 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
       rw [h'', natDegree_zero] at h'
       exact Nat.not_succ_le_zero 0 h'
     use normalizeScaleRoots_monic p this
-    rw [normalizeScaleRoots_eval₂_leadingCoeff_mul p h' f x, h, MulZeroClass.mul_zero]
+    rw [normalizeScaleRoots_eval₂_leadingCoeff_mul p h' f x, h, mul_zero]
   · by_cases hp : p.map f = 0
     · apply_fun fun q => coeff q p.natDegree at hp
       rw [coeff_map, coeff_zero, coeff_natDegree] at hp
-      rw [hp, MulZeroClass.zero_mul]
+      rw [hp, zero_mul]
       exact f.is_integral_zero
     · rw [Nat.one_le_iff_ne_zero, Classical.not_not] at h'
       rw [eq_C_of_natDegree_eq_zero h', eval₂_C] at h

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

This has nice performance benefits.

@@ -37,7 +37,7 @@ open Classical BigOperators Polynomial Submodule
 
 section Ring
 
-variable {R S A : Type _}
+variable {R S A : Type*}
 
 variable [CommRing R] [Ring A] [Ring S] (f : R →+* S)
 
@@ -115,7 +115,7 @@ end Ring
 
 section
 
-variable {K A : Type _}
+variable {K A : Type*}
 
 variable [Field K] [Ring A] [Algebra K A] [FiniteDimensional K A]
 
@@ -135,13 +135,13 @@ end
 
 section
 
-variable {R A B S : Type _}
+variable {R A B S : Type*}
 
 variable [CommRing R] [CommRing A] [CommRing B] [CommRing S]
 
 variable [Algebra R A] [Algebra R B] (f : R →+* S)
 
-theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
+theorem map_isIntegral {B C F : Type*} [Ring B] [Ring C] [Algebra R B] [Algebra A B] [Algebra R C]
     [IsScalarTower R A B] [Algebra A C] [IsScalarTower R A C] {b : B} [AlgHomClass F A B C] (f : F)
     (hb : IsIntegral R b) : IsIntegral R (f b) := by
   obtain ⟨P, hP⟩ := hb
@@ -150,7 +150,7 @@ theorem map_isIntegral {B C F : Type _} [Ring B] [Ring C] [Algebra R B] [Algebra
     aeval_algHom_apply, aeval_map_algebraMap, aeval_def, hP.2, _root_.map_zero]
 #align map_is_integral map_isIntegral
 
-theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R] [CommRing S]
+theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type*} [CommRing R] [CommRing S]
     [CommRing T] [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
     (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
     IsIntegral T (ψ a) := by
@@ -161,7 +161,7 @@ theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R]
     RingHom.map_zero]
 #align is_integral_map_of_comp_eq_of_is_integral isIntegral_map_of_comp_eq_of_isIntegral
 
-theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
+theorem isIntegral_algHom_iff {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A →ₐ[R] B) (hf : Function.Injective f) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x := by
   refine' ⟨_, map_isIntegral f⟩
   rintro ⟨p, hp, hx⟩
@@ -171,7 +171,7 @@ theorem isIntegral_algHom_iff {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Al
 #align is_integral_alg_hom_iff isIntegral_algHom_iff
 
 @[simp]
-theorem isIntegral_algEquiv {A B : Type _} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
+theorem isIntegral_algEquiv {A B : Type*} [Ring A] [Ring B] [Algebra R A] [Algebra R B]
     (f : A ≃ₐ[R] B) {x : A} : IsIntegral R (f x) ↔ IsIntegral R x :=
   ⟨fun h => by simpa using map_isIntegral f.symm.toAlgHom h, map_isIntegral f.toAlgHom⟩
 #align is_integral_alg_equiv isIntegral_algEquiv
@@ -182,7 +182,7 @@ theorem isIntegral_of_isScalarTower [Algebra A B] [IsScalarTower R A B] {x : B}
   ⟨p.map <| algebraMap R A, hp.map _, by rw [← aeval_def, aeval_map_algebraMap, aeval_def, hpx]⟩
 #align is_integral_of_is_scalar_tower isIntegral_of_isScalarTower
 
-theorem map_isIntegral_int {B C F : Type _} [Ring B] [Ring C] {b : B} [RingHomClass F B C] (f : F)
+theorem map_isIntegral_int {B C F : Type*} [Ring B] [Ring C] {b : B} [RingHomClass F B C] (f : F)
     (hb : IsIntegral ℤ b) : IsIntegral ℤ (f b) :=
   map_isIntegral (f : B →+* C).toIntAlgHom hb
 #align map_is_integral_int map_isIntegral_int
@@ -369,14 +369,14 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
   exact Subalgebra.smul_mem _ (Algebra.subset_adjoin <| hlx1 hr) _
 #align is_integral_of_mem_of_fg isIntegral_of_mem_of_FG
 
-theorem Module.End.isIntegral {M : Type _} [AddCommGroup M] [Module R M] [Module.Finite R M] :
+theorem Module.End.isIntegral {M : Type*} [AddCommGroup M] [Module R M] [Module.Finite R M] :
     Algebra.IsIntegral R (Module.End R M) :=
   LinearMap.exists_monic_and_aeval_eq_zero R
 #align module.End.is_integral Module.End.isIntegral
 
 /-- Suppose `A` is an `R`-algebra, `M` is an `A`-module such that `a • m ≠ 0` for all non-zero `a`
 and `m`. If `x : A` fixes a nontrivial f.g. `R`-submodule `N` of `M`, then `x` is `R`-integral. -/
-theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R M] [Module A M]
+theorem isIntegral_of_smul_mem_submodule {M : Type*} [AddCommGroup M] [Module R M] [Module A M]
     [IsScalarTower R A M] [NoZeroSMulDivisors A M] (N : Submodule R M) (hN : N ≠ ⊥) (hN' : N.FG)
     (x : A) (hx : ∀ n ∈ N, x • n ∈ N) : IsIntegral R x := by
   let A' : Subalgebra R A :=
@@ -643,7 +643,7 @@ theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x
     (fun _ => isIntegral_neg) fun _ _ => isIntegral_mul
 #align is_integral_of_mem_closure' isIntegral_of_mem_closure'
 
-theorem isIntegral_of_mem_closure'' {S : Type _} [CommRing S] {f : R →+* S} (G : Set S)
+theorem isIntegral_of_mem_closure'' {S : Type*} [CommRing S] {f : R →+* S} (G : Set S)
     (hG : ∀ x ∈ G, f.IsIntegralElem x) : ∀ x ∈ Subring.closure G, f.IsIntegralElem x := fun x hx =>
   @isIntegral_of_mem_closure' R S _ _ f.toAlgebra G hG x hx
 #align is_integral_of_mem_closure'' isIntegral_of_mem_closure''
@@ -670,17 +670,17 @@ theorem IsIntegral.multiset_sum {s : Multiset A} (h : ∀ x ∈ s, IsIntegral R
   (integralClosure R A).multiset_sum_mem h
 #align is_integral.multiset_sum IsIntegral.multiset_sum
 
-theorem IsIntegral.prod {α : Type _} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
+theorem IsIntegral.prod {α : Type*} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
     IsIntegral R (∏ x in s, f x) :=
   (integralClosure R A).prod_mem h
 #align is_integral.prod IsIntegral.prod
 
-theorem IsIntegral.sum {α : Type _} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
+theorem IsIntegral.sum {α : Type*} {s : Finset α} (f : α → A) (h : ∀ x ∈ s, IsIntegral R (f x)) :
     IsIntegral R (∑ x in s, f x) :=
   (integralClosure R A).sum_mem h
 #align is_integral.sum IsIntegral.sum
 
-theorem IsIntegral.det {n : Type _} [Fintype n] [DecidableEq n] {M : Matrix n n A}
+theorem IsIntegral.det {n : Type*} [Fintype n] [DecidableEq n] {M : Matrix n n A}
     (h : ∀ i j, IsIntegral R (M i j)) : IsIntegral R M.det := by
   rw [Matrix.det_apply]
   exact IsIntegral.sum _ fun σ _hσ => IsIntegral.zsmul (IsIntegral.prod _ fun i _hi => h _ _) _
@@ -832,13 +832,13 @@ section IsIntegralClosure
 the integral closure of `R` in `B`,
 i.e. that an element of `B` is integral over `R` iff it is an element of (the image of) `A`.
 -/
-class IsIntegralClosure (A R B : Type _) [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B]
+class IsIntegralClosure (A R B : Type*) [CommRing R] [CommSemiring A] [CommRing B] [Algebra R B]
   [Algebra A B] : Prop where
   algebraMap_injective' : Function.Injective (algebraMap A B)
   isIntegral_iff : ∀ {x : B}, IsIntegral R x ↔ ∃ y, algebraMap A B y = x
 #align is_integral_closure IsIntegralClosure
 
-instance integralClosure.isIntegralClosure (R A : Type _) [CommRing R] [CommRing A] [Algebra R A] :
+instance integralClosure.isIntegralClosure (R A : Type*) [CommRing R] [CommRing A] [Algebra R A] :
     IsIntegralClosure (integralClosure R A) R A where
   algebraMap_injective' := Subtype.coe_injective
   isIntegral_iff {x} := ⟨fun h => ⟨⟨x, h⟩, rfl⟩, by rintro ⟨⟨_, h⟩, rfl⟩; exact h⟩
@@ -847,11 +847,11 @@ instance integralClosure.isIntegralClosure (R A : Type _) [CommRing R] [CommRing
 namespace IsIntegralClosure
 
 -- Porting note: added to work around missing infer kind support
-theorem algebraMap_injective (A R B : Type _) [CommRing R] [CommSemiring A] [CommRing B]
+theorem algebraMap_injective (A R B : Type*) [CommRing R] [CommSemiring A] [CommRing B]
   [Algebra R B] [Algebra A B] [IsIntegralClosure A R B] : Function.Injective (algebraMap A B) :=
   algebraMap_injective' R
 
-variable {R A B : Type _} [CommRing R] [CommRing A] [CommRing B]
+variable {R A B : Type*} [CommRing R] [CommRing A] [CommRing B]
 
 variable [Algebra R B] [Algebra A B] [IsIntegralClosure A R B]
 
@@ -918,7 +918,7 @@ theorem mk'_algebraMap [Algebra R A] [IsScalarTower R A B] (x : R)
 section lift
 
 -- porting note: `R` and `A` were redundant binder updates
-variable (B) {S : Type _} [CommRing S] [Algebra R S]
+variable (B) {S : Type*} [CommRing S] [Algebra R S]
 -- split from above, since otherwise it does not synthesize `Semiring S`
 variable [Algebra S B] [IsScalarTower R S B]
 
@@ -944,7 +944,7 @@ end lift
 
 section Equiv
 
-variable (R B) (A' : Type _) [CommRing A']
+variable (R B) (A' : Type*) [CommRing A']
 variable [Algebra A' B] [IsIntegralClosure A' R B]
 
 variable [Algebra R A] [Algebra R A'] [IsScalarTower R A B] [IsScalarTower R A' B]
@@ -977,7 +977,7 @@ section Algebra
 
 open Algebra
 
-variable {R A B S T : Type _}
+variable {R A B S T : Type*}
 
 variable [CommRing R] [CommRing A] [CommRing B] [CommRing S] [CommRing T]
 
@@ -1071,7 +1071,7 @@ nonrec theorem RingHom.isIntegral_tower_bot_of_isIntegral (hg : Function.Injecti
     hg x (hfg (g x))
 #align ring_hom.is_integral_tower_bot_of_is_integral RingHom.isIntegral_tower_bot_of_isIntegral
 
-theorem isIntegral_tower_bot_of_isIntegral_field {R A B : Type _} [CommRing R] [Field A]
+theorem isIntegral_tower_bot_of_isIntegral_field {R A B : Type*} [CommRing R] [Field A]
     [CommRing B] [Nontrivial B] [Algebra R A] [Algebra A B] [Algebra R B] [IsScalarTower R A B]
     {x : A} (h : IsIntegral R (algebraMap A B x)) : IsIntegral R x :=
   isIntegral_tower_bot_of_isIntegral (algebraMap A B).injective h
@@ -1123,7 +1123,7 @@ theorem isIntegral_quotientMap_iff {I : Ideal S} :
 #align is_integral_quotient_map_iff isIntegral_quotientMap_iff
 
 /-- If the integral extension `R → S` is injective, and `S` is a field, then `R` is also a field. -/
-theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial R] [CommRing S]
+theorem isField_of_isIntegral_of_isField {R S : Type*} [CommRing R] [Nontrivial R] [CommRing S]
     [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S)
     (hRS : Function.Injective (algebraMap R S)) (hS : IsField S) : IsField R := by
   refine' ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun {a} ha => _⟩
@@ -1161,7 +1161,7 @@ theorem isField_of_isIntegral_of_isField {R S : Type _} [CommRing R] [Nontrivial
   rw [mul_assoc, ← pow_succ', tsub_add_cancel_of_le this]
 #align is_field_of_is_integral_of_is_field isField_of_isIntegral_of_isField
 
-theorem isField_of_isIntegral_of_isField' {R S : Type _} [CommRing R] [CommRing S] [IsDomain S]
+theorem isField_of_isIntegral_of_isField' {R S : Type*} [CommRing R] [CommRing S] [IsDomain S]
     [Algebra R S] (H : Algebra.IsIntegral R S) (hR : IsField R) : IsField S := by
   letI := hR.toField
   refine' ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun {x} hx => _⟩
@@ -1178,7 +1178,7 @@ theorem isField_of_isIntegral_of_isField' {R S : Type _} [CommRing R] [CommRing
   exact ⟨y, Subtype.ext_iff.mp hy⟩
 #align is_field_of_is_integral_of_is_field' isField_of_isIntegral_of_isField'
 
-theorem Algebra.IsIntegral.isField_iff_isField {R S : Type _} [CommRing R] [Nontrivial R]
+theorem Algebra.IsIntegral.isField_iff_isField {R S : Type*} [CommRing R] [Nontrivial R]
     [CommRing S] [IsDomain S] [Algebra R S] (H : Algebra.IsIntegral R S)
     (hRS : Function.Injective (algebraMap R S)) : IsField R ↔ IsField S :=
   ⟨isField_of_isIntegral_of_isField' H, isField_of_isIntegral_of_isField H hRS⟩
@@ -1186,7 +1186,7 @@ theorem Algebra.IsIntegral.isField_iff_isField {R S : Type _} [CommRing R] [Nont
 
 end Algebra
 
-theorem integralClosure_idem {R : Type _} {A : Type _} [CommRing R] [CommRing A] [Algebra R A] :
+theorem integralClosure_idem {R : Type*} {A : Type*} [CommRing R] [CommRing A] [Algebra R A] :
     integralClosure (integralClosure R A : Set A) A = ⊥ :=
   eq_bot_iff.2 fun x hx =>
     Algebra.mem_bot.2
@@ -1198,7 +1198,7 @@ theorem integralClosure_idem {R : Type _} {A : Type _} [CommRing R] [CommRing A]
 
 section IsDomain
 
-variable {R S : Type _} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S]
+variable {R S : Type*} [CommRing R] [CommRing S] [IsDomain S] [Algebra R S]
 
 instance : IsDomain (integralClosure R S) :=
   inferInstance

This:

• Improves the module docstring, which was both out of date and not very informative
• Addresses a TODO to generalize includeLeft to commuting actions. As a result a few downstream results are changed to be about includeLeftRingHom or a ⊗ₜ 1, as carrying around the extra useless ring just makes the lemmas harder to use. Nothing seems to suffer from this change.
• Introduces TensorProduct.AlgebraTensorModule.rid
• Generalizes the following to work for towers of rings:
  • Algebra.TensorProduct.algHomOfLinearMapTensorProduct
  • Algebra.TensorProduct.map
  • Algebra.TensorProduct.congr
  • Algebra.TensorProduct.endTensorEndAlgHom
  • Algebra.TensorProduct.ext (and renames it to Algebra.TensorProduct.ext')
  • Algebra.TensorProduct.rid
• Introduces a new Algebra.TensorProduct.ext which follows "partially-applied ext lemmas", and uses it to golf a proof in RingTheory/Etale.lean

I need many of these results for building AlgEquivs relating to the base change of clifford algebras.

@@ -708,7 +708,7 @@ theorem IsIntegral.tmul (x : A) {y : B} (h : IsIntegral R y) : IsIntegral A (x 
   · simp only [Algebra.TensorProduct.includeLeftRingHom_apply, Algebra.TensorProduct.tmul_pow,
       one_pow]
     convert (MulZeroClass.mul_zero (M₀ := A ⊗[R] B) _).symm
-    erw [Polynomial.eval₂_map, Algebra.TensorProduct.includeLeft_comp_algebraMap,
+    erw [Polynomial.eval₂_map, Algebra.TensorProduct.includeLeftRingHom_comp_algebraMap,
       ← Polynomial.eval₂_map]
     convert Polynomial.eval₂_at_apply
       (Algebra.TensorProduct.includeRight : B →ₐ[R] A ⊗[R] B).toRingHom y

Also

• fix the names of minpoly.natDegree_le and minpoly.degree_le
• rename minpoly.ne_zero_of_finite_field_extension to minpoly.ne_zero_of_finite
• reduce typeclass assumptions of some lemmas in RingTheory/Algebraic
• add two lemmas isIntegral_of_finite and isAlgebraic_of_finite
• move Algebra.isIntegral_of_finite to RingTheory/IntegralClosure

@@ -115,6 +115,26 @@ end Ring
 
 section
 
+variable {K A : Type _}
+
+variable [Field K] [Ring A] [Algebra K A] [FiniteDimensional K A]
+
+variable (K)
+
+theorem isIntegral_of_finite (e : A) : IsIntegral K e :=
+  isIntegral_of_noetherian (IsNoetherian.iff_fg.2 inferInstance) _
+
+variable (A)
+
+/-- A field extension is integral if it is finite. -/
+theorem Algebra.isIntegral_of_finite : Algebra.IsIntegral K A := fun x =>
+  isIntegral_of_submodule_noetherian ⊤ (IsNoetherian.iff_fg.2 inferInstance) x Algebra.mem_top
+#align algebra.is_integral_of_finite Algebra.isIntegral_of_finite
+
+end
+
+section
+
 variable {R A B S : Type _}
 
 variable [CommRing R] [CommRing A] [CommRing B] [CommRing S]

• depends on: #5966

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

@@ -2,11 +2,6 @@
 Copyright (c) 2019 Kenny Lau. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Kenny Lau
-
-! This file was ported from Lean 3 source module ring_theory.integral_closure
-! leanprover-community/mathlib commit 641b6a82006416ec431b2987b354af9311fed4f2
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Data.Polynomial.Expand
 import Mathlib.LinearAlgebra.FiniteDimensional
@@ -17,6 +12,8 @@ import Mathlib.RingTheory.Polynomial.ScaleRoots
 import Mathlib.RingTheory.Polynomial.Tower
 import Mathlib.RingTheory.TensorProduct
 
+#align_import ring_theory.integral_closure from "leanprover-community/mathlib"@"641b6a82006416ec431b2987b354af9311fed4f2"
+
 /-!
 # Integral closure of a subring.
 

Co-authored-by: Jason Yuen <jason_yuen2007@hotmail.com> Co-authored-by: int-y1 <jason_yuen2007@hotmail.com> Co-authored-by: Antoine Chambert-Loir <antoine.chambert-loir@math.univ-paris-diderot.fr> Co-authored-by: Johan Commelin <johan@commelin.net> Co-authored-by: Jujian Zhang <jujian.zhang1998@outlook.com> Co-authored-by: Parcly Taxel <reddeloostw@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com>

@@ -606,17 +606,17 @@ theorem integralClosure.isIntegral (x : integralClosure R A) : IsIntegral R x :=
       rwa [← aeval_def, ← Subalgebra.val_apply, aeval_algHom_apply] at hpx⟩
 #align integral_closure.is_integral integralClosure.isIntegral
 
-theorem RingHom.is_integral_of_is_integral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
+theorem RingHom.isIntegral_of_isIntegral_mul_unit (x y : S) (r : R) (hr : f r * y = 1)
     (hx : f.IsIntegralElem (x * y)) : f.IsIntegralElem x := by
   obtain ⟨p, ⟨p_monic, hp⟩⟩ := hx
   refine' ⟨scaleRoots p r, ⟨(monic_scaleRoots_iff r).2 p_monic, _⟩⟩
   convert scaleRoots_eval₂_eq_zero f hp
   rw [mul_comm x y, ← mul_assoc, hr, one_mul]
-#align ring_hom.is_integral_of_is_integral_mul_unit RingHom.is_integral_of_is_integral_mul_unit
+#align ring_hom.is_integral_of_is_integral_mul_unit RingHom.isIntegral_of_isIntegral_mul_unit
 
 theorem isIntegral_of_isIntegral_mul_unit {x y : A} {r : R} (hr : algebraMap R A r * y = 1)
     (hx : IsIntegral R (x * y)) : IsIntegral R x :=
-  (algebraMap R A).is_integral_of_is_integral_mul_unit x y r hr hx
+  (algebraMap R A).isIntegral_of_isIntegral_mul_unit x y r hr hx
 #align is_integral_of_is_integral_mul_unit isIntegral_of_isIntegral_mul_unit
 
 /-- Generalization of `isIntegral_of_mem_closure` bootstrapped up from that lemma -/
@@ -626,10 +626,10 @@ theorem isIntegral_of_mem_closure' (G : Set A) (hG : ∀ x ∈ G, IsIntegral R x
     (fun _ => isIntegral_neg) fun _ _ => isIntegral_mul
 #align is_integral_of_mem_closure' isIntegral_of_mem_closure'
 
-theorem is_integral_of_mem_closure'' {S : Type _} [CommRing S] {f : R →+* S} (G : Set S)
+theorem isIntegral_of_mem_closure'' {S : Type _} [CommRing S] {f : R →+* S} (G : Set S)
     (hG : ∀ x ∈ G, f.IsIntegralElem x) : ∀ x ∈ Subring.closure G, f.IsIntegralElem x := fun x hx =>
   @isIntegral_of_mem_closure' R S _ _ f.toAlgebra G hG x hx
-#align is_integral_of_mem_closure'' is_integral_of_mem_closure''
+#align is_integral_of_mem_closure'' isIntegral_of_mem_closure''
 
 theorem IsIntegral.pow {x : A} (h : IsIntegral R x) (n : ℕ) : IsIntegral R (x ^ n) :=
   (integralClosure R A).pow_mem h n

The result of running

find . -type f -name "*.lean" -exec sed -i -E 'N;s/^#align ([^[:space:]]+)\n *([^[:space:]]+)$/#align \1 \2/' {} \;

Hopefully for the last time...

Co-authored-by: Moritz Firsching <firsching@google.com>

@@ -720,8 +720,7 @@ theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ nat
     apply Nat.le_pred_of_lt
     rw [lt_iff_le_and_ne]
     exact ⟨le_natDegree_of_ne_zero h₁, h₂⟩
-#align normalize_scale_roots_coeff_mul_leading_coeff_pow
-  normalizeScaleRoots_coeff_mul_leadingCoeff_pow
+#align normalize_scale_roots_coeff_mul_leading_coeff_pow normalizeScaleRoots_coeff_mul_leadingCoeff_pow
 
 theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
     p.leadingCoeff • normalizeScaleRoots p = scaleRoots p p.leadingCoeff := by

Currently, (for both Set and Finset) insert_subset is an iff lemma stating that insert a s ⊆ t if and only if a ∈ t and s ⊆ t. For both types, this PR renames this lemma to insert_subset_iff, and adds an insert_subset lemma that gives the implication just in the reverse direction : namely theorem insert_subset (ha : a ∈ t) (hs : s ⊆ t) : insert a s ⊆ t .

This both aligns the naming with union_subset and union_subset_iff, and removes the need for the awkward insert_subset.mpr ⟨_,_⟩ idiom. It touches a lot of files (too many to list), but in a trivial way.

@@ -321,7 +321,8 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
     constructor <;> intro hz
     · exact
         (span_le.2
-          (Set.insert_subset.2 ⟨(Algebra.adjoin S₀ (y : Set A)).one_mem, Algebra.subset_adjoin⟩)) hz
+          (Set.insert_subset_iff.2
+            ⟨(Algebra.adjoin S₀ (y : Set A)).one_mem, Algebra.subset_adjoin⟩)) hz
     · rw [Subalgebra.mem_toSubmodule, Algebra.mem_adjoin_iff] at hz
       suffices Subring.closure (Set.range (algebraMap (↥S₀) A) ∪ ↑y) ≤ S₁ by exact this hz
       refine' Subring.closure_le.2 (Set.union_subset _ fun t ht => subset_span <| Or.inr ht)
@@ -719,7 +720,8 @@ theorem normalizeScaleRoots_coeff_mul_leadingCoeff_pow (i : ℕ) (hp : 1 ≤ nat
     apply Nat.le_pred_of_lt
     rw [lt_iff_le_and_ne]
     exact ⟨le_natDegree_of_ne_zero h₁, h₂⟩
-#align normalize_scale_roots_coeff_mul_leading_coeff_pow normalizeScaleRoots_coeff_mul_leadingCoeff_pow
+#align normalize_scale_roots_coeff_mul_leading_coeff_pow
+  normalizeScaleRoots_coeff_mul_leadingCoeff_pow
 
 theorem leadingCoeff_smul_normalizeScaleRoots (p : R[X]) :
     p.leadingCoeff • normalizeScaleRoots p = scaleRoots p p.leadingCoeff := by
@@ -1063,7 +1065,8 @@ theorem RingHom.isIntegralElem_of_isIntegralElem_comp {x : T} (h : (g.comp f).Is
     g.IsIntegralElem x :=
   let ⟨p, ⟨hp, hp'⟩⟩ := h
   ⟨p.map f, hp.map f, by rwa [← eval₂_map] at hp'⟩
-#align ring_hom.is_integral_elem_of_is_integral_elem_comp RingHom.isIntegralElem_of_isIntegralElem_comp
+#align ring_hom.is_integral_elem_of_is_integral_elem_comp
+  RingHom.isIntegralElem_of_isIntegralElem_comp
 
 theorem RingHom.isIntegral_tower_top_of_isIntegral (h : (g.comp f).IsIntegral) : g.IsIntegral :=
   fun x => RingHom.isIntegralElem_of_isIntegralElem_comp f g (h x)

Changes are of the form

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

@@ -137,7 +137,7 @@ theorem isIntegral_map_of_comp_eq_of_isIntegral {R S T U : Type _} [CommRing R]
     [CommRing T] [CommRing U] [Algebra R S] [Algebra T U] (φ : R →+* T) (ψ : S →+* U)
     (h : (algebraMap T U).comp φ = ψ.comp (algebraMap R S)) {a : S} (ha : IsIntegral R a) :
     IsIntegral T (ψ a) := by
-  rw [IsIntegral, RingHom.IsIntegralElem] at ha⊢
+  rw [IsIntegral, RingHom.IsIntegralElem] at ha ⊢
   obtain ⟨p, hp⟩ := ha
   refine' ⟨p.map φ, hp.left.map _, _⟩
   rw [← eval_map, map_map, h, ← map_map, eval_map, eval₂_at_apply, eval_map, hp.right,
@@ -215,7 +215,7 @@ theorem FG_adjoin_singleton_of_integral (x : A) (hx : IsIntegral R x) :
   rw [← aeval_def] at hfx
   rw [AlgHom.map_add, AlgHom.map_mul, hfx, MulZeroClass.zero_mul, add_zero]
   have : degree (p %ₘ f) ≤ degree f := degree_modByMonic_le p hfm
-  generalize p %ₘ f = q at this⊢
+  generalize p %ₘ f = q at this ⊢
   rw [← sum_C_mul_X_pow_eq q, aeval_def, eval₂_sum, sum_def]
   refine' sum_mem fun k hkq => _
   rw [eval₂_mul, eval₂_C, eval₂_pow, eval₂_X, ← Algebra.smul_def]
@@ -384,7 +384,7 @@ theorem isIntegral_of_smul_mem_submodule {M : Type _} [AddCommGroup M] [Module R
       (by intros x y; ext; apply mul_smul)
   obtain ⟨a, ha₁, ha₂⟩ : ∃ a ∈ N, a ≠ (0 : M) := by
     by_contra h'
-    push_neg  at h'
+    push_neg at h'
     apply hN
     rwa [eq_bot_iff]
   have : Function.Injective f := by
@@ -785,7 +785,7 @@ theorem RingHom.isIntegralElem_leadingCoeff_mul (h : p.eval₂ f x = 0) :
     use normalizeScaleRoots_monic p this
     rw [normalizeScaleRoots_eval₂_leadingCoeff_mul p h' f x, h, MulZeroClass.mul_zero]
   · by_cases hp : p.map f = 0
-    · apply_fun fun q => coeff q p.natDegree  at hp
+    · apply_fun fun q => coeff q p.natDegree at hp
       rw [coeff_map, coeff_zero, coeff_natDegree] at hp
       rw [hp, MulZeroClass.zero_mul]
       exact f.is_integral_zero

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Mario Carneiro <di.gama@gmail.com> Co-authored-by: Floris van Doorn <fpvdoorn@gmail.com> Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Alex J Best <alex.j.best@gmail.com>

@@ -436,16 +436,18 @@ theorem Algebra.IsIntegral.finite (h : Algebra.IsIntegral R A) [h' : Algebra.Fin
         -- Porting note: was `ext`
         refine IsScalarTower.Algebra.ext (algebraMap R A).toAlgebra _ fun r x => ?_
         exact (Algebra.smul_def _ _).symm)
-  -- porting note: the rest of the proof was
-  -- `delta RingHom.Finite at this; convert this; ext; exact Algebra.smul_def _ _`
-  rw [RingHom.Finite] at this; convert this; ext; rfl
+  rw [RingHom.Finite] at this
+  convert this
+  ext
+  exact Algebra.smul_def _ _
 #align algebra.is_integral.finite Algebra.IsIntegral.finite
 
 theorem Algebra.IsIntegral.of_finite [h : Module.Finite R A] : Algebra.IsIntegral R A := by
   apply RingHom.Finite.to_isIntegral
-  -- porting note: the rest of the proof was
-  -- `delta RingHom.Finite; convert h; ext; exact (Algebra.smul_def _ _).symm`
-  rw [RingHom.Finite]; convert h; ext; rfl
+  rw [RingHom.Finite]
+  convert h
+  ext
+  exact (Algebra.smul_def _ _).symm
 #align algebra.is_integral.of_finite Algebra.IsIntegral.of_finite
 
 /-- finite = integral + finite type -/

Currently, the following notations are changed from · ×ˢ · because Lean 4 can't deal with ambiguous notations. | Definition | Notation | | :

Co-authored-by: Jeremy Tan Jie Rui <reddeloostw@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: Chris Hughes <chrishughes24@gmail.com>

@@ -266,7 +266,8 @@ theorem isIntegral_of_mem_of_FG (S : Subalgebra R A) (HS : S.toSubmodule.FG) (x
       rw [← hy]
       exact subset_span hp
   -- Now `S` is a subalgebra so the product of two elements of `y` is also in `S`.
-  have : ∀ jk : (y ×ᶠ y : Set (A × A)), jk.1.1 * jk.1.2 ∈ (Subalgebra.toSubmodule S) := fun jk =>
+  have : ∀ jk : (y ×ˢ y : Finset (A × A)),
+      jk.1.1 * jk.1.2 ∈ (Subalgebra.toSubmodule S) := fun jk =>
     S.mul_mem (hyS (Finset.mem_product.1 jk.2).1) (hyS (Finset.mem_product.1 jk.2).2)
   rw [← hy, ← Set.image_id (y : Set A)] at this
   simp only [Finsupp.mem_span_image_iff_total] at this

I tried to fix what I could, but this contains a lot of sorries!

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>